GCSE Maths Standard Form, Indices and Surds (Foundation): worked examples and questions with answers

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Count how many times each number is multiplied; that count is the index.

1. Count how many times each number appears. The index of a power counts the number of equal factors in the multiplication.
   5 appears 4 times
2. Write each number with its count as the index. A number that appears c times is written as (number)^c.
   5 appears 4 times: 5⁴
3. Write the answer.
   5⁴

Check. Expand it again: 5⁴ = 5 × 5 × 5 × 5; the same list of factors. ✓

Watch out. The index is the number of lots, not the answer. 5 × 5 × 5 is 5³ (=125), not 5 × 3 (=15).

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Count how many times each number is multiplied; that count is the index.

1. Count how many times each number appears. The index of a power counts the number of equal factors in the multiplication.
   2 appears 6 times
2. Write each number with its count as the index. A number that appears c times is written as (number)^c.
   2 appears 6 times: 2⁶
3. Write the answer.
   2⁶

Check. Expand it again: 2⁶ = 2 × 2 × 2 × 2 × 2 × 2; the same list of factors. ✓

Watch out. The index is the number of lots, not the answer. 5 × 5 × 5 is 5³ (=125), not 5 × 3 (=15).

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Count how many times each number is multiplied; that count is the index.

1. Count how many times each number appears. The index of a power counts the number of equal factors in the multiplication.
   10 appears 3 times
2. Write each number with its count as the index. A number that appears c times is written as (number)^c.
   10 appears 3 times: 10³
3. Write the answer.
   10³

Check. Expand it again: 10³ = 10 × 10 × 10; the same list of factors. ✓

Watch out. The index is the number of lots, not the answer. 5 × 5 × 5 is 5³ (=125), not 5 × 3 (=15).

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Count how many times each number is multiplied; that count is the index.

1. Count how many times each number appears. The index of a power counts the number of equal factors in the multiplication.
   7 appears 2 times
2. Write each number with its count as the index. A number that appears c times is written as (number)^c.
   7 appears 2 times: 7²
3. Write the answer.
   7²

Check. Expand it again: 7² = 7 × 7; the same list of factors. ✓

Watch out. The index is the number of lots, not the answer. 5 × 5 × 5 is 5³ (=125), not 5 × 3 (=15).

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the power as repeated multiplication of the base, then multiply.

1. Say what the power means. The index 5 tells us to multiply 5 lots of 2 together.
   2⁵ = 2 × 2 × 2 × 2 × 2
2. Multiply step by step, left to right. Do one multiplication at a time so it is easy to follow.
   2 × 2 = 4
   4 × 2 = 8
   8 × 2 = 16
   16 × 2 = 32

Check. Split the power: 2² × 2³ = 4 × 8 = 32. ✓

Watch out. 2⁵ does not mean 2 × 5. The index says how many lots of 2 are multiplied.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the power as repeated multiplication of the base, then multiply.

1. Say what the power means. The index 4 tells us to multiply 4 lots of 3 together.
   3⁴ = 3 × 3 × 3 × 3
2. Multiply step by step, left to right. Do one multiplication at a time so it is easy to follow.
   3 × 3 = 9
   9 × 3 = 27
   27 × 3 = 81

Check. Split the power: 3² × 3² = 9 × 9 = 81. ✓

Watch out. 3⁴ does not mean 3 × 4. The index says how many lots of 3 are multiplied.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the power as repeated multiplication of the base, then multiply.

1. Say what the power means. The index 3 tells us to multiply 3 lots of 10 together.
   10³ = 10 × 10 × 10
2. Multiply step by step, left to right. Do one multiplication at a time so it is easy to follow.
   10 × 10 = 100
   100 × 10 = 1000

Check. Split the power: 10¹ × 10² = 10 × 100 = 1000. ✓

Watch out. 10³ does not mean 10 × 3. The index says how many lots of 10 are multiplied.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the power as repeated multiplication of the base, then multiply.

1. Say what the power means. The index 3 tells us to multiply 3 lots of 5 together.
   5³ = 5 × 5 × 5
2. Multiply step by step, left to right. Do one multiplication at a time so it is easy to follow.
   5 × 5 = 25
   25 × 5 = 125

Check. Split the power: 5¹ × 5² = 5 × 25 = 125. ✓

Watch out. 5³ does not mean 5 × 3. The index says how many lots of 5 are multiplied.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the power as repeated multiplication of the base, then multiply.

1. Say what the power means. The index 3 tells us to multiply 3 lots of 4 together.
   4³ = 4 × 4 × 4
2. Multiply step by step, left to right. Do one multiplication at a time so it is easy to follow.
   4 × 4 = 16
   16 × 4 = 64

Check. Split the power: 4¹ × 4² = 4 × 16 = 64. ✓

Watch out. 4³ does not mean 4 × 3. The index says how many lots of 4 are multiplied.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the power as repeated multiplication of the base, then multiply.

1. Say what the power means. The index 2 tells us to multiply 2 lots of 6 together.
   6² = 6 × 6
2. Multiply step by step, left to right. Do one multiplication at a time so it is easy to follow.
   6 × 6 = 36

Watch out. 6² does not mean 6 × 2. The index says how many lots of 6 are multiplied.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the power as repeated multiplication of the base, then multiply.

1. Say what the power means. The index 1 tells us to multiply 1 lots of 2 together.
   2¹ = 2
2. Write the answer. An index of 1 means just one lot of the base, so the number does not change.
   2¹ = 2

Watch out. 2¹ does not mean 2 × 1. The index says how many lots of 2 are multiplied.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the power as repeated multiplication of the base, then multiply.

1. Say what the power means. The index 2 tells us to multiply 2 lots of 2 together.
   2² = 2 × 2
2. Multiply step by step, left to right. Do one multiplication at a time so it is easy to follow.
   2 × 2 = 4

Watch out. The index says how many lots are multiplied; it is not the same as multiplying the base by the index.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the power as repeated multiplication of the base, then multiply.

1. Say what the power means. The index 3 tells us to multiply 3 lots of 2 together.
   2³ = 2 × 2 × 2
2. Multiply step by step, left to right. Do one multiplication at a time so it is easy to follow.
   2 × 2 = 4
   4 × 2 = 8

Check. Split the power: 2¹ × 2² = 2 × 4 = 8. ✓

Watch out. 2³ does not mean 2 × 3. The index says how many lots of 2 are multiplied.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the power as repeated multiplication of the base, then multiply.

1. Say what the power means. The index 4 tells us to multiply 4 lots of 2 together.
   2⁴ = 2 × 2 × 2 × 2
2. Multiply step by step, left to right. Do one multiplication at a time so it is easy to follow.
   2 × 2 = 4
   4 × 2 = 8
   8 × 2 = 16

Check. Split the power: 2² × 2² = 4 × 4 = 16. ✓

Watch out. 2⁴ does not mean 2 × 4. The index says how many lots of 2 are multiplied.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the power as repeated multiplication of the base, then multiply.

1. Say what the power means. The index 5 tells us to multiply 5 lots of 2 together.
   2⁵ = 2 × 2 × 2 × 2 × 2
2. Multiply step by step, left to right. Do one multiplication at a time so it is easy to follow.
   2 × 2 = 4
   4 × 2 = 8
   8 × 2 = 16
   16 × 2 = 32

Check. Split the power: 2² × 2³ = 4 × 8 = 32. ✓

Watch out. 2⁵ does not mean 2 × 5. The index says how many lots of 2 are multiplied.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. When you multiply powers with the same base, you add the indices: a^m × aⁿ = a^m+n.

1. Check that the bases are the same. These laws only work when the base (the big number) is the same in both powers.
   Both bases are 2.
2. Write each power out in full. This shows what is really happening: we are counting how many lots of the base are multiplied.
   2³ = 2 × 2 × 2
   2⁴ = 2 × 2 × 2 × 2
   Together: 3 lots of 2 and 4 lots of 2 make 7 lots of 2.
3. Add the indices. Joining the lots means adding the number of lots.
   3 + 4 = 7
4. Write the answer as a single power. Keep the same base and use the new index.
   2³ × 2⁴ = 2⁷

Check. Work out the numbers: 2³ = 8 and 2⁴ = 16. 8 × 16 = 128, and 2⁷ = 128. ✓

Watch out. Add the indices when you multiply. Do not multiply the indices, and do not multiply the bases (3² × 3⁴ is not 9⁶).

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. When you multiply powers with the same base, you add the indices: a^m × aⁿ = a^m+n.

1. Check that the bases are the same. These laws only work when the base (the big number) is the same in both powers.
   Both bases are 5.
2. Write each power out in full. This shows what is really happening: we are counting how many lots of the base are multiplied.
   5² = 5 × 5
   5⁶ = 5 × 5 × 5 × 5 × 5 × 5
   Together: 2 lots of 5 and 6 lots of 5 make 8 lots of 5.
3. Add the indices. Joining the lots means adding the number of lots.
   2 + 6 = 8
4. Write the answer as a single power. Keep the same base and use the new index.
   5² × 5⁶ = 5⁸

Check. Work out the numbers: 5² = 25 and 5⁶ = 15,625. 25 × 15,625 = 390,625, and 5⁸ = 390,625. ✓

Watch out. Add the indices when you multiply. Do not multiply the indices, and do not multiply the bases (3² × 3⁴ is not 9⁶).

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. When you multiply powers with the same base, you add the indices: a^m × aⁿ = a^m+n.

1. Check that the bases are the same. These laws only work when the base (the big number) is the same in both powers.
   Both bases are 7.
2. Write each power out in full. This shows what is really happening: we are counting how many lots of the base are multiplied.
   7⁵ = 7 × 7 × 7 × 7 × 7
   7¹ = 7
   Together: 5 lots of 7 and 1 lots of 7 make 6 lots of 7.
3. Add the indices. Joining the lots means adding the number of lots.
   5 + 1 = 6
4. Write the answer as a single power. Keep the same base and use the new index.
   7⁵ × 7¹ = 7⁶

Check. Work out the numbers: 7⁵ = 16,807 and 7¹ = 7. 16,807 × 7 = 117,649, and 7⁶ = 117,649. ✓

Watch out. Add the indices when you multiply. Do not multiply the indices, and do not multiply the bases (3² × 3⁴ is not 9⁶).

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. When you multiply powers with the same base, you add the indices: a^m × aⁿ = a^m+n.

1. Check that the bases are the same. These laws only work when the base (the big number) is the same in both powers.
   Both bases are 10.
2. Write each power out in full. This shows what is really happening: we are counting how many lots of the base are multiplied.
   10³ = 10 × 10 × 10
   10³ = 10 × 10 × 10
   Together: 3 lots of 10 and 3 lots of 10 make 6 lots of 10.
3. Add the indices. Joining the lots means adding the number of lots.
   3 + 3 = 6
4. Write the answer as a single power. Keep the same base and use the new index.
   10³ × 10³ = 10⁶

Check. Work out the numbers: 10³ = 1000 and 10³ = 1000. 1000 × 1000 = 1,000,000, and 10⁶ = 1,000,000. ✓

Watch out. Add the indices when you multiply. Do not multiply the indices, and do not multiply the bases (3² × 3⁴ is not 9⁶).

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. When you divide powers with the same base, you subtract the indices: a^m ÷ aⁿ = a^m−n.

1. Check that the bases are the same. These laws only work when the base (the big number) is the same in both powers.
   Both bases are 6.
2. Say what the rule means. Dividing cancels lots of the base from the top: every lot on the bottom removes one lot from the top.
   6⁸ has 8 lots of 6; we take away 3 lots.
3. Subtract the indices. The number of lots left is the first index take away the second.
   8 − 3 = 5
4. Write the answer as a single power. Keep the same base and use the new index.
   6⁸ ÷ 6³ = 6⁵

Check. Work out the numbers: 6⁸ = 1,679,616 and 6³ = 216. 1,679,616 ÷ 216 = 7776, and 6⁵ = 7776. ✓

Watch out. Subtract the indices when you divide. Do not divide the indices, and never divide the bases.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. When you divide powers with the same base, you subtract the indices: a^m ÷ aⁿ = a^m−n.

1. Check that the bases are the same. These laws only work when the base (the big number) is the same in both powers.
   Both bases are 4.
2. Say what the rule means. Dividing cancels lots of the base from the top: every lot on the bottom removes one lot from the top.
   4⁹ has 9 lots of 4; we take away 5 lots.
3. Subtract the indices. The number of lots left is the first index take away the second.
   9 − 5 = 4
4. Write the answer as a single power. Keep the same base and use the new index.
   4⁹ ÷ 4⁵ = 4⁴

Check. Work out the numbers: 4⁹ = 262,144 and 4⁵ = 1024. 262,144 ÷ 1024 = 256, and 4⁴ = 256. ✓

Watch out. Subtract the indices when you divide. Do not divide the indices, and never divide the bases.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. When you divide powers with the same base, you subtract the indices: a^m ÷ aⁿ = a^m−n.

1. Check that the bases are the same. These laws only work when the base (the big number) is the same in both powers.
   Both bases are 3.
2. Write both powers out in full as a fraction. Dividing is the same as writing a fraction, and then we can cancel (divide top and bottom by the same thing).
   3 × 3 × 3 × 3 × 3 × 3 × 33 × 3
   Each 3 on the bottom cancels with one 3 on the top (3 ÷ 3 = 1).
   2 of the 7 are cancelled, leaving 5 lots of 3.
3. Subtract the indices. The number of lots left is the first index take away the second.
   7 − 2 = 5
4. Write the answer as a single power. Keep the same base and use the new index.
   3⁷ ÷ 3² = 3⁵

Check. Work out the numbers: 3⁷ = 2187 and 3² = 9. 2187 ÷ 9 = 243, and 3⁵ = 243. ✓

Watch out. Subtract the indices when you divide. Do not divide the indices, and never divide the bases.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. When you divide powers with the same base, you subtract the indices: a^m ÷ aⁿ = a^m−n.

1. Check that the bases are the same. These laws only work when the base (the big number) is the same in both powers.
   Both bases are 10.
2. Write both powers out in full as a fraction. Dividing is the same as writing a fraction, and then we can cancel (divide top and bottom by the same thing).
   10 × 10 × 10 × 10 × 10 × 1010 × 10 × 10 × 10
   Each 10 on the bottom cancels with one 10 on the top (10 ÷ 10 = 1).
   4 of the 6 are cancelled, leaving 2 lots of 10.
3. Subtract the indices. The number of lots left is the first index take away the second.
   6 − 4 = 2
4. Write the answer as a single power. Keep the same base and use the new index.
   10⁶ ÷ 10⁴ = 10²

Check. Work out the numbers: 10⁶ = 1,000,000 and 10⁴ = 10,000. 1,000,000 ÷ 10,000 = 100, and 10² = 100. ✓

Watch out. Subtract the indices when you divide. Do not divide the indices, and never divide the bases.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. To raise a power to a power, multiply the indices: (a^m)ⁿ = a^m×n.

1. Write the bracket out as repeated multiplication. The outer index 4 means 4 lots of 3² multiplied together.
   (3²)⁴ = 3² × 3² × 3² × 3²
   = 3^{2 + 2 + 2 + 2}
   That is 4 lots of 2, which is 4 × 2 = 8.
2. Multiply the indices. Each of the outer lots holds m of the base, so the total is m × n.
   2 × 4 = 8
3. Write the answer as a single power. Keep the same base and use the new index.
   (3²)⁴ = 3⁸

Check. Work out the numbers: 3² = 9, and 9⁴ = 6561. Also 3⁸ = 6561. ✓

Watch out. A power of a power: multiply the indices. Adding them would be for 3² × 3⁴, which is a different question.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. To raise a power to a power, multiply the indices: (a^m)ⁿ = a^m×n.

1. Write the bracket out as repeated multiplication. The outer index 2 means 2 lots of 5³ multiplied together.
   (5³)² = 5³ × 5³
   = 5^{3 + 3}
   That is 2 lots of 3, which is 2 × 3 = 6.
2. Multiply the indices. Each of the outer lots holds m of the base, so the total is m × n.
   3 × 2 = 6
3. Write the answer as a single power. Keep the same base and use the new index.
   (5³)² = 5⁶

Check. Work out the numbers: 5³ = 125, and 125² = 15,625. Also 5⁶ = 15,625. ✓

Watch out. A power of a power: multiply the indices. Adding them would be for 3² × 3⁴, which is a different question.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. To raise a power to a power, multiply the indices: (a^m)ⁿ = a^m×n.

1. Write the bracket out as repeated multiplication. The outer index 3 means 3 lots of 2⁵ multiplied together.
   (2⁵)³ = 2⁵ × 2⁵ × 2⁵
   = 2^{5 + 5 + 5}
   That is 3 lots of 5, which is 3 × 5 = 15.
2. Multiply the indices. Each of the outer lots holds m of the base, so the total is m × n.
   5 × 3 = 15
3. Write the answer as a single power. Keep the same base and use the new index.
   (2⁵)³ = 2¹⁵

Check. Work out the numbers: 2⁵ = 32, and 32³ = 32,768. Also 2¹⁵ = 32,768. ✓

Watch out. A power of a power: multiply the indices. Adding them would be for 3² × 3⁴, which is a different question.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. To raise a power to a power, multiply the indices: (a^m)ⁿ = a^m×n.

1. Write the bracket out as repeated multiplication. The outer index 3 means 3 lots of 10² multiplied together.
   (10²)³ = 10² × 10² × 10²
   = 10^{2 + 2 + 2}
   That is 3 lots of 2, which is 3 × 2 = 6.
2. Multiply the indices. Each of the outer lots holds m of the base, so the total is m × n.
   2 × 3 = 6
3. Write the answer as a single power. Keep the same base and use the new index.
   (10²)³ = 10⁶

Check. Work out the numbers: 10² = 100, and 100³ = 1,000,000. Also 10⁶ = 1,000,000. ✓

Watch out. A power of a power: multiply the indices. Adding them would be for 3² × 3⁴, which is a different question.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. When you multiply powers with the same base, you add the indices: a^m × aⁿ = a^m+n.

1. Check that the bases are the same. These laws only work when the base (the big number) is the same in both powers.
   Both bases are 2.
2. Write each power out in full. This shows what is really happening: we are counting how many lots of the base are multiplied.
   2³ = 2 × 2 × 2
   2² = 2 × 2
   Together: 3 lots of 2 and 2 lots of 2 make 5 lots of 2.
3. Add the indices. Joining the lots means adding the number of lots.
   3 + 2 = 5
4. Write the answer as a single power. Keep the same base and use the new index.
   2³ × 2² = 2⁵
5. Work out the value. 2⁵ means 5 lots of 2 multiplied together.
   2⁵ = 2 × 2 × 2 × 2 × 2 = 32

Check. Work out the numbers: 2³ = 8 and 2² = 4. 8 × 4 = 32, and 2⁵ = 32. ✓

Watch out. Add the indices when you multiply. Do not multiply the indices, and do not multiply the bases (3² × 3⁴ is not 9⁶).

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. When you divide powers with the same base, you subtract the indices: a^m ÷ aⁿ = a^m−n.

1. Check that the bases are the same. These laws only work when the base (the big number) is the same in both powers.
   Both bases are 3.
2. Write both powers out in full as a fraction. Dividing is the same as writing a fraction, and then we can cancel (divide top and bottom by the same thing).
   3 × 3 × 3 × 3 × 33 × 3 × 3
   Each 3 on the bottom cancels with one 3 on the top (3 ÷ 3 = 1).
   3 of the 5 are cancelled, leaving 2 lots of 3.
3. Subtract the indices. The number of lots left is the first index take away the second.
   5 − 3 = 2
4. Write the answer as a single power. Keep the same base and use the new index.
   3⁵ ÷ 3³ = 3²
5. Work out the value. 3² means 2 lots of 3 multiplied together.
   3² = 3 × 3 = 9

Check. Work out the numbers: 3⁵ = 243 and 3³ = 27. 243 ÷ 27 = 9, and 3² = 9. ✓

Watch out. Subtract the indices when you divide. Do not divide the indices, and never divide the bases.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. To raise a power to a power, multiply the indices: (a^m)ⁿ = a^m×n.

1. Write the bracket out as repeated multiplication. The outer index 3 means 3 lots of 2² multiplied together.
   (2²)³ = 2² × 2² × 2²
   = 2^{2 + 2 + 2}
   That is 3 lots of 2, which is 3 × 2 = 6.
2. Multiply the indices. Each of the outer lots holds m of the base, so the total is m × n.
   2 × 3 = 6
3. Write the answer as a single power. Keep the same base and use the new index.
   (2²)³ = 2⁶
4. Work out the value. 2⁶ means 6 lots of 2 multiplied together.
   2⁶ = 2 × 2 × 2 × 2 × 2 × 2 = 64

Check. Work out the numbers: 2² = 4, and 4³ = 64. Also 2⁶ = 64. ✓

Watch out. A power of a power: multiply the indices. Adding them would be for 3² × 3⁴, which is a different question.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. When you multiply powers with the same base, you add the indices: a^m × aⁿ = a^m+n.

1. Check that the bases are the same. These laws only work when the base (the big number) is the same in both powers.
   Both bases are 10.
2. Write each power out in full. This shows what is really happening: we are counting how many lots of the base are multiplied.
   10² = 10 × 10
   10³ = 10 × 10 × 10
   Together: 2 lots of 10 and 3 lots of 10 make 5 lots of 10.
3. Add the indices. Joining the lots means adding the number of lots.
   2 + 3 = 5
4. Write the answer as a single power. Keep the same base and use the new index.
   10² × 10³ = 10⁵
5. Work out the value. 10⁵ means 5 lots of 10 multiplied together.
   10⁵ = 10 × 10 × 10 × 10 × 10 = 100,000

Check. Work out the numbers: 10² = 100 and 10³ = 1000. 100 × 1000 = 100,000, and 10⁵ = 100,000. ✓

Watch out. Add the indices when you multiply. Do not multiply the indices, and do not multiply the bases (3² × 3⁴ is not 9⁶).

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. When you divide powers with the same base, you subtract the indices: a^m ÷ aⁿ = a^m−n.

1. Check that the bases are the same. These laws only work when the base (the big number) is the same in both powers.
   Both bases are 5.
2. Write both powers out in full as a fraction. Dividing is the same as writing a fraction, and then we can cancel (divide top and bottom by the same thing).
   5 × 5 × 5 × 55 × 5
   Each 5 on the bottom cancels with one 5 on the top (5 ÷ 5 = 1).
   2 of the 4 are cancelled, leaving 2 lots of 5.
3. Subtract the indices. The number of lots left is the first index take away the second.
   4 − 2 = 2
4. Write the answer as a single power. Keep the same base and use the new index.
   5⁴ ÷ 5² = 5²
5. Work out the value. 5² means 2 lots of 5 multiplied together.
   5² = 5 × 5 = 25

Check. Work out the numbers: 5⁴ = 625 and 5² = 25. 625 ÷ 25 = 25, and 5² = 25. ✓

Watch out. Subtract the indices when you divide. Do not divide the indices, and never divide the bases.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Work out each power first, then multiply the results.

1. Work out each power separately. A power means repeated multiplication of the base.
   2³ = 2 × 2 × 2 = 8
   5² = 5 × 5 = 25
2. Multiply the values together. The original question was a product of the powers.
   8 × 25 = 200

Check. Multiplying in a different order gives the same result: 25 × 8 = 200. ✓

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Work out each power first, then multiply the results.

1. Work out each power separately. A power means repeated multiplication of the base.
   2² = 2 × 2 = 4
   3³ = 3 × 3 × 3 = 27
2. Multiply the values together. The original question was a product of the powers.
   4 × 27 = 108

Check. Multiplying in a different order gives the same result: 27 × 4 = 108. ✓

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Work out each power first, then multiply the results.

1. Work out each power separately. A power means repeated multiplication of the base.
   10³ = 10 × 10 × 10 = 1000
   4² = 4 × 4 = 16
2. Multiply the values together. The original question was a product of the powers.
   1000 × 16 = 16,000

Check. Multiplying in a different order gives the same result: 16 × 1000 = 16,000. ✓

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Work out each power first, then multiply the results.

1. Work out each power separately. A power means repeated multiplication of the base.
   3² = 3 × 3 = 9
   2⁴ = 2 × 2 × 2 × 2 = 16
2. Multiply the values together. The original question was a product of the powers.
   9 × 16 = 144

Check. Multiplying in a different order gives the same result: 16 × 9 = 144. ✓

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Use the law of indices to write an equation for the missing index.

1. Check that the bases match. The laws of indices need the same base on both sides.
   The base is 3 throughout.
2. Write an equation for the indices. Multiplying powers adds the indices.
   4 + ? = 9
3. Solve it for the missing index. Use the inverse operation.
   ? = 9 − 4 = 5
   ? = 5

Check. 3⁴ × 3⁵ = 3^{4 + 5} = 3⁹. ✓

Watch out. Check by putting your answer back into the question.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Use the law of indices to write an equation for the missing index.

1. Check that the bases match. The laws of indices need the same base on both sides.
   The base is 7 throughout.
2. Write an equation for the indices. Multiplying powers adds the indices.
   ? + 2 = 5
3. Solve it for the missing index. Use the inverse operation.
   ? = 5 − 2 = 3
   ? = 3

Check. 7³ × 7² = 7^{3 + 2} = 7⁵. ✓

Watch out. Check by putting your answer back into the question.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Use the law of indices to write an equation for the missing index.

1. Check that the bases match. The laws of indices need the same base on both sides.
   The base is 5 throughout.
2. Write an equation for the indices. Dividing powers subtracts the indices.
   8 − ? = 3
3. Solve it for the missing index. Use the inverse operation.
   ? = 8 − 3 = 5
   ? = 5

Check. 5⁸ ÷ 5⁵ = 5^{8 − 5} = 5³. ✓

Watch out. Check by putting your answer back into the question.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Use the law of indices to write an equation for the missing index.

1. Check that the bases match. The laws of indices need the same base on both sides.
   The base is 2 throughout.
2. Write an equation for the indices. Dividing powers subtracts the indices.
   ? − 3 = 4
3. Solve it for the missing index. Use the inverse operation.
   ? = 4 + 3 = 7
   ? = 7

Check. 2⁷ ÷ 2³ = 2^{7 − 3} = 2⁴. ✓

Watch out. Check by putting your answer back into the question.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Use the law of indices to write an equation for the missing index.

1. Check that the bases match. The laws of indices need the same base on both sides.
   The base is 4 throughout.
2. Write an equation for the indices. A power of a power multiplies the indices.
   ? × 3 = 12
3. Solve it for the missing index. Use the inverse operation.
   ? = 12 ÷ 3 = 4
   ? = 4

Check. (4⁴)³ = 4^{4 × 3} = 4¹². ✓

Watch out. Check by putting your answer back into the question.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Use the laws of indices in order: add indices to multiply, subtract indices to divide.

1. Check that every base is the same. The laws of indices only work when the bases match.
   Every base is 5.
2. Work out the top first: multiply, so add the indices. a^m × aⁿ = a^m+n.
   7 + 2 = 9, so the top is 5⁹
3. Divide: subtract the indices. a^m ÷ aⁿ = a^m−n.
   5⁹5⁶ = 5⁹ ÷ 5⁶ = 5³, because 9 − 6 = 3

Check. The total of the indices on top is 9 and on the bottom is 6; 9 − 6 = 3. ✓

Watch out. Deal with the multiplication in the top (and bottom) first, then divide. Only subtract once the top and bottom are each a single power.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Use the laws of indices in order: add indices to multiply, subtract indices to divide.

1. Check that every base is the same. The laws of indices only work when the bases match.
   Every base is 2.
2. Work out the top first: multiply, so add the indices. a^m × aⁿ = a^m+n.
   3 + 5 = 8, so the top is 2⁸
3. Divide: subtract the indices. a^m ÷ aⁿ = a^m−n.
   2⁸2⁴ = 2⁸ ÷ 2⁴ = 2⁴, because 8 − 4 = 4

Check. The total of the indices on top is 8 and on the bottom is 4; 8 − 4 = 4. ✓

Watch out. Deal with the multiplication in the top (and bottom) first, then divide. Only subtract once the top and bottom are each a single power.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Use the laws of indices in order: add indices to multiply, subtract indices to divide.

1. Check that every base is the same. The laws of indices only work when the bases match.
   Every base is 3.
2. Work out the top first: multiply, so add the indices. a^m × aⁿ = a^m+n.
   9 = 9, so the top is 3⁹
3. Work out the bottom: add the indices. The bottom is also a multiplication of the same base.
   2 + 3 = 5, so the bottom is 3⁵
4. Divide: subtract the indices. a^m ÷ aⁿ = a^m−n.
   3⁹3⁵ = 3⁹ ÷ 3⁵ = 3⁴, because 9 − 5 = 4

Check. The total of the indices on top is 9 and on the bottom is 5; 9 − 5 = 4. ✓

Watch out. Deal with the multiplication in the top (and bottom) first, then divide. Only subtract once the top and bottom are each a single power.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Use the laws of indices in order: add indices to multiply, subtract indices to divide.

1. Check that every base is the same. The laws of indices only work when the bases match.
   Every base is 7.
2. Work out the top first: multiply, so add the indices. a^m × aⁿ = a^m+n.
   4 + 3 = 7, so the top is 7⁷
3. Work out the bottom: add the indices. The bottom is also a multiplication of the same base.
   2 + 1 = 3, so the bottom is 7³
4. Divide: subtract the indices. a^m ÷ aⁿ = a^m−n.
   7⁷7³ = 7⁷ ÷ 7³ = 7⁴, because 7 − 3 = 4

Check. The total of the indices on top is 7 and on the bottom is 3; 7 − 3 = 4. ✓

Watch out. Deal with the multiplication in the top (and bottom) first, then divide. Only subtract once the top and bottom are each a single power.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. After each fold the layers double, so after 6 folds there are 2 × 2 × 2 × 2 × 2 × 2 layers.

1. Write it as a power. Six doublings means six lots of 2 multiplied together.
   2⁶
2. Multiply out. Work one step at a time.
   2 × 2 = 4, 4 × 2 = 8, 8 × 2 = 16, 16 × 2 = 32, 32 × 2 = 64
3. Write the answer.
   2⁶ = 64 layers

Check. Fold 3 times to get 8, and fold 3 more times: 8 × 8 = 64. ✓

Watch out. After 6 folds the index is 6, not the number of layers.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. List the powers of 2 until one is bigger than 1000.

1. Write the powers of 2. Each fold doubles the layers, so n folds give 2ⁿ layers.
   2⁶ = 64, 2⁷ = 128, 2⁸ = 256, 2⁹ = 512, 2¹⁰ = 1024
2. Find the first one above 1000. 2⁹ = 512 is too small. 2¹⁰ = 1024 is bigger than 1000.
   The smallest number of folds is 10.

Check. 9 folds give 512 layers (not enough) and 10 folds give 1024 layers (more than 1000). ✓

Watch out. You need strictly more than 1000, so 512 is not enough.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the power as repeated multiplication of the base, then multiply.

1. Say what the power means. The index 5 tells us to multiply 5 lots of 3 together.
   3⁵ = 3 × 3 × 3 × 3 × 3
2. Multiply step by step, left to right. Do one multiplication at a time so it is easy to follow.
   3 × 3 = 9
   9 × 3 = 27
   27 × 3 = 81
   81 × 3 = 243

Check. Split the power: 3² × 3³ = 9 × 27 = 243. ✓

Watch out. 3⁵ does not mean 3 × 5. The index says how many lots of 3 are multiplied.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Work out the numbers in each round and multiply, then also use the index law.

1. Round 3 and round 2. 3³ people and 3² people.
   3³ = 27 and 3² = 9
2. Multiply them.
   27 × 9 = 243
3. Compare with round 5. The index law says 3³ × 3² = 3³⁺² = 3⁵.
   3⁵ = 243 (from the first part), so they are equal.

Check. 27 × 9 = 243 and 3⁵ = 243. ✓

Watch out. Add the indices (3 + 2 = 5), do not multiply them (3 × 2 = 6).

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Multiply the numbers, then add the indices of each letter: a^m × aⁿ = a^m+n.

1. Deal with the letter x: add the indices. The letter is the same in both terms, so the index law applies. (x on its own means x¹.)
   x³ × x⁴ = x^{3 + 4} = x⁷
2. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   x⁷

Check. Choose numbers: x = 2. The original expression has the value 128, and the answer x⁷ also has the value 128. ✓

Watch out. Add the indices of the same letter only. You cannot combine different letters: a² × b³ stays as it is.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Multiply the numbers, then add the indices of each letter: a^m × aⁿ = a^m+n.

1. Deal with the letter y: add the indices. The letter is the same in both terms, so the index law applies. (y on its own means y¹.)
   y⁵ × y² = y^{5 + 2} = y⁷
2. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   y⁷

Check. Choose numbers: y = 3. The original expression has the value 2187, and the answer y⁷ also has the value 2187. ✓

Watch out. Add the indices of the same letter only. You cannot combine different letters: a² × b³ stays as it is.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Multiply the numbers, then add the indices of each letter: a^m × aⁿ = a^m+n.

1. Deal with the letter a: add the indices. The letter is the same in both terms, so the index law applies. (a on its own means a¹.)
   a¹ × a⁶ = a^{1 + 6} = a⁷
2. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   a⁷

Check. Choose numbers: a = 2. The original expression has the value 128, and the answer a⁷ also has the value 128. ✓

Watch out. Add the indices of the same letter only. You cannot combine different letters: a² × b³ stays as it is.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Multiply the numbers, then add the indices of each letter: a^m × aⁿ = a^m+n.

1. Deal with the letter p: add the indices. The letter is the same in both terms, so the index law applies. (p on its own means p¹.)
   p³ × p³ = p^{3 + 3} = p⁶
2. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   p⁶

Check. Choose numbers: p = 2. The original expression has the value 64, and the answer p⁶ also has the value 64. ✓

Watch out. Add the indices of the same letter only. You cannot combine different letters: a² × b³ stays as it is.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Divide the numbers, then subtract the indices of each letter: a^m ÷ aⁿ = a^m−n.

1. Deal with the letter x: subtract the indices. The letter is the same in both terms, so the index law applies. (x on its own means x¹.)
   x⁹ ÷ x⁴ = x^{9 − 4} = x⁵
2. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   x⁵

Check. Choose numbers: x = 2. The original expression has the value 32, and the answer x⁵ also has the value 32. ✓

Watch out. Subtract the indices (top minus bottom), and divide the numbers. Do not subtract the numbers.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Divide the numbers, then subtract the indices of each letter: a^m ÷ aⁿ = a^m−n.

1. Deal with the letter y: subtract the indices. The letter is the same in both terms, so the index law applies. (y on its own means y¹.)
   y⁷ ÷ y² = y^{7 − 2} = y⁵
2. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   y⁵

Check. Choose numbers: y = 3. The original expression has the value 243, and the answer y⁵ also has the value 243. ✓

Watch out. Subtract the indices (top minus bottom), and divide the numbers. Do not subtract the numbers.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Divide the numbers, then subtract the indices of each letter: a^m ÷ aⁿ = a^m−n.

1. Deal with the letter m: subtract the indices. The letter is the same in both terms, so the index law applies. (m on its own means m¹.)
   m⁸ ÷ m¹ = m^{8 − 1} = m⁷
2. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   m⁷

Check. Choose numbers: m = 2. The original expression has the value 128, and the answer m⁷ also has the value 128. ✓

Watch out. Subtract the indices (top minus bottom), and divide the numbers. Do not subtract the numbers.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Divide the numbers, then subtract the indices of each letter: a^m ÷ aⁿ = a^m−n.

1. Deal with the letter t: subtract the indices. The letter is the same in both terms, so the index law applies. (t on its own means t¹.)
   t⁵ ÷ t⁴ = t^{5 − 4} = t¹
2. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   t

Check. Choose numbers: t = 2. The original expression has the value 2, and the answer t also has the value 2. ✓

Watch out. Subtract the indices (top minus bottom), and divide the numbers. Do not subtract the numbers.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Raise the number and every letter to the power. Multiply the indices: (a^m)ⁿ = a^m×n.

1. Raise the letter x to the power: multiply the indices. (a^m)ⁿ = a^m×n.
   (x³)² = x^{3 × 2} = x⁶
2. Put it all together.
   x⁶

Check. Choose numbers: x = 2. The original expression has the value 64, and the answer x⁶ also has the value 64. ✓

Watch out. The number in front is raised to the power too: (2a)³ = 8a³, not 2a³.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Raise the number and every letter to the power. Multiply the indices: (a^m)ⁿ = a^m×n.

1. Raise the letter y to the power: multiply the indices. (a^m)ⁿ = a^m×n.
   (y⁴)³ = y^{4 × 3} = y¹²
2. Put it all together.
   y¹²

Check. Choose numbers: y = 3. The original expression has the value 531,441, and the answer y¹² also has the value 531,441. ✓

Watch out. The number in front is raised to the power too: (2a)³ = 8a³, not 2a³.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Raise the number and every letter to the power. Multiply the indices: (a^m)ⁿ = a^m×n.

1. Raise the letter a to the power: multiply the indices. (a^m)ⁿ = a^m×n.
   (a²)⁵ = a^{2 × 5} = a¹⁰
2. Put it all together.
   a¹⁰

Check. Choose numbers: a = 2. The original expression has the value 1024, and the answer a¹⁰ also has the value 1024. ✓

Watch out. The number in front is raised to the power too: (2a)³ = 8a³, not 2a³.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Raise the number and every letter to the power. Multiply the indices: (a^m)ⁿ = a^m×n.

1. Raise the letter p to the power: multiply the indices. (a^m)ⁿ = a^m×n.
   (p⁶)² = p^{6 × 2} = p¹²
2. Put it all together.
   p¹²

Check. Choose numbers: p = 2. The original expression has the value 4096, and the answer p¹² also has the value 4096. ✓

Watch out. The number in front is raised to the power too: (2a)³ = 8a³, not 2a³.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Multiply the numbers, then add the indices of each letter: a^m × aⁿ = a^m+n.

1. Multiply the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.
   3 × 4 = 12
2. Deal with the letter x: add the indices. The letter is the same in both terms, so the index law applies. (x on its own means x¹.)
   x² × x⁵ = x^{2 + 5} = x⁷
3. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   12x⁷

Check. Choose numbers: x = 2. The original expression has the value 1536, and the answer 12x⁷ also has the value 1536. ✓

Watch out. Add the indices of the same letter only. You cannot combine different letters: a² × b³ stays as it is.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Multiply the numbers, then add the indices of each letter: a^m × aⁿ = a^m+n.

1. Multiply the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.
   2 × 5 = 10
2. Deal with the letter y: add the indices. The letter is the same in both terms, so the index law applies. (y on its own means y¹.)
   y³ × y¹ = y^{3 + 1} = y⁴
3. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   10y⁴

Check. Choose numbers: y = 3. The original expression has the value 810, and the answer 10y⁴ also has the value 810. ✓

Watch out. Add the indices of the same letter only. You cannot combine different letters: a² × b³ stays as it is.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Multiply the numbers, then add the indices of each letter: a^m × aⁿ = a^m+n.

1. Multiply the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.
   6 × 2 = 12
2. Deal with the letter a: add the indices. The letter is the same in both terms, so the index law applies. (a on its own means a¹.)
   a⁴ × a² = a^{4 + 2} = a⁶
3. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   12a⁶

Check. Choose numbers: a = 2. The original expression has the value 768, and the answer 12a⁶ also has the value 768. ✓

Watch out. Add the indices of the same letter only. You cannot combine different letters: a² × b³ stays as it is.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Multiply the numbers, then add the indices of each letter: a^m × aⁿ = a^m+n.

1. Multiply the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.
   7 × 3 = 21
2. Deal with the letter p: add the indices. The letter is the same in both terms, so the index law applies. (p on its own means p¹.)
   p¹ × p³ = p^{1 + 3} = p⁴
3. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   21p⁴

Check. Choose numbers: p = 2. The original expression has the value 336, and the answer 21p⁴ also has the value 336. ✓

Watch out. Add the indices of the same letter only. You cannot combine different letters: a² × b³ stays as it is.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Divide the numbers, then subtract the indices of each letter: a^m ÷ aⁿ = a^m−n.

1. Divide the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.
   12 ÷ 4 = 3
2. Deal with the letter y: subtract the indices. The letter is the same in both terms, so the index law applies. (y on its own means y¹.)
   y⁸ ÷ y³ = y^{8 − 3} = y⁵
3. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   3y⁵

Check. Choose numbers: y = 3. The original expression has the value 729, and the answer 3y⁵ also has the value 729. ✓

Watch out. Subtract the indices (top minus bottom), and divide the numbers. Do not subtract the numbers.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Divide the numbers, then subtract the indices of each letter: a^m ÷ aⁿ = a^m−n.

1. Divide the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.
   20 ÷ 5 = 4
2. Deal with the letter a: subtract the indices. The letter is the same in both terms, so the index law applies. (a on its own means a¹.)
   a⁷ ÷ a² = a^{7 − 2} = a⁵
3. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   4a⁵

Check. Choose numbers: a = 2. The original expression has the value 128, and the answer 4a⁵ also has the value 128. ✓

Watch out. Subtract the indices (top minus bottom), and divide the numbers. Do not subtract the numbers.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Divide the numbers, then subtract the indices of each letter: a^m ÷ aⁿ = a^m−n.

1. Divide the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.
   18 ÷ 6 = 3
2. Deal with the letter x: subtract the indices. The letter is the same in both terms, so the index law applies. (x on its own means x¹.)
   x⁵ ÷ x¹ = x^{5 − 1} = x⁴
3. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   3x⁴

Check. Choose numbers: x = 2. The original expression has the value 48, and the answer 3x⁴ also has the value 48. ✓

Watch out. Subtract the indices (top minus bottom), and divide the numbers. Do not subtract the numbers.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Divide the numbers, then subtract the indices of each letter: a^m ÷ aⁿ = a^m−n.

1. Divide the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.
   8 ÷ 2 = 4
2. Deal with the letter m: subtract the indices. The letter is the same in both terms, so the index law applies. (m on its own means m¹.)
   m⁶ ÷ m⁶ = m^{6 − 6} = 1 (anything to the power 0 is 1)
3. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   4

Check. Choose numbers: m = 2. The original expression has the value 4, and the answer 4 also has the value 4. ✓

Watch out. Subtract the indices (top minus bottom), and divide the numbers. Do not subtract the numbers.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Raise the number and every letter to the power. Multiply the indices: (a^m)ⁿ = a^m×n.

1. Raise the number in front to the power. Everything inside the bracket is raised to the power, and that includes the number.
   (2)⁴ = 2 × 2 × 2 × 2 = 16
2. Raise the letter a to the power: multiply the indices. (a^m)ⁿ = a^m×n.
   (a³)⁴ = a^{3 × 4} = a¹²
3. Put it all together.
   16a¹²

Check. Choose numbers: a = 2. The original expression has the value 65,536, and the answer 16a¹² also has the value 65,536. ✓

Watch out. The number in front is raised to the power too: (2a)³ = 8a³, not 2a³.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Raise the number and every letter to the power. Multiply the indices: (a^m)ⁿ = a^m×n.

1. Raise the number in front to the power. Everything inside the bracket is raised to the power, and that includes the number.
   (5)³ = 5 × 5 × 5 = 125
2. Raise the letter x to the power: multiply the indices. (a^m)ⁿ = a^m×n.
   (x²)³ = x^{2 × 3} = x⁶
3. Put it all together.
   125x⁶

Check. Choose numbers: x = 2. The original expression has the value 8000, and the answer 125x⁶ also has the value 8000. ✓

Watch out. The number in front is raised to the power too: (2a)³ = 8a³, not 2a³.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Raise the number and every letter to the power. Multiply the indices: (a^m)ⁿ = a^m×n.

1. Raise the number in front to the power. Everything inside the bracket is raised to the power, and that includes the number.
   (3)² = 3 × 3 = 9
2. Raise the letter y to the power: multiply the indices. (a^m)ⁿ = a^m×n.
   (y⁴)² = y^{4 × 2} = y⁸
3. Put it all together.
   9y⁸

Check. Choose numbers: y = 3. The original expression has the value 59,049, and the answer 9y⁸ also has the value 59,049. ✓

Watch out. The number in front is raised to the power too: (2a)³ = 8a³, not 2a³.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Raise the number and every letter to the power. Multiply the indices: (a^m)ⁿ = a^m×n.

1. Raise the number in front to the power. Everything inside the bracket is raised to the power, and that includes the number.
   (10)² = 10 × 10 = 100
2. Raise the letter p to the power: multiply the indices. (a^m)ⁿ = a^m×n.
   (p³)² = p^{3 × 2} = p⁶
3. Put it all together.
   100p⁶

Check. Choose numbers: p = 2. The original expression has the value 6400, and the answer 100p⁶ also has the value 6400. ✓

Watch out. The number in front is raised to the power too: (2a)³ = 8a³, not 2a³.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Multiply the numbers, then add the indices of each letter: a^m × aⁿ = a^m+n.

1. Multiply the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.
   3 × 5 = 15
2. Deal with the letter a: add the indices. The letter is the same in both terms, so the index law applies. (a on its own means a¹.)
   a² × a¹ = a^{2 + 1} = a³
3. Deal with the letter b: add the indices. The letter is the same in both terms, so the index law applies. (b on its own means b¹.)
   b¹ × b³ = b^{1 + 3} = b⁴
4. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   15a³b⁴

Check. Choose numbers: a = 2, b = 3. The original expression has the value 9720, and the answer 15a³b⁴ also has the value 9720. ✓

Watch out. Add the indices of the same letter only. You cannot combine different letters: a² × b³ stays as it is.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Multiply the numbers, then add the indices of each letter: a^m × aⁿ = a^m+n.

1. Multiply the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.
   4 × 2 = 8
2. Deal with the letter x: add the indices. The letter is the same in both terms, so the index law applies. (x on its own means x¹.)
   x² × x¹ = x^{2 + 1} = x³
3. Deal with the letter y: add the indices. The letter is the same in both terms, so the index law applies. (y on its own means y¹.)
   y¹ × y² = y^{1 + 2} = y³
4. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   8x³y³

Check. Choose numbers: x = 2, y = 3. The original expression has the value 1728, and the answer 8x³y³ also has the value 1728. ✓

Watch out. Add the indices of the same letter only. You cannot combine different letters: a² × b³ stays as it is.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Divide the numbers, then subtract the indices of each letter: a^m ÷ aⁿ = a^m−n.

1. Divide the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.
   6 ÷ 3 = 2
2. Deal with the letter p: subtract the indices. The letter is the same in both terms, so the index law applies. (p on its own means p¹.)
   p³ ÷ p¹ = p^{3 − 1} = p²
3. Deal with the letter q: subtract the indices. The letter is the same in both terms, so the index law applies. (q on its own means q¹.)
   q² ÷ q¹ = q^{2 − 1} = q¹
4. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   2p²q

Check. Choose numbers: p = 2, q = 3. The original expression has the value 24, and the answer 2p²q also has the value 24. ✓

Watch out. Subtract the indices (top minus bottom), and divide the numbers. Do not subtract the numbers.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x³ the coefficient is 7. A letter on its own, like x, really means x¹.

Method. Divide the numbers, then subtract the indices of each letter: a^m ÷ aⁿ = a^m−n.

1. Divide the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.
   10 ÷ 5 = 2
2. Deal with the letter m: subtract the indices. The letter is the same in both terms, so the index law applies. (m on its own means m¹.)
   m⁵ ÷ m² = m^{5 − 2} = m³
3. Deal with the letter n: subtract the indices. The letter is the same in both terms, so the index law applies. (n on its own means n¹.)
   n⁴ ÷ n¹ = n^{4 − 1} = n³
4. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   2m³n³

Check. Choose numbers: m = 2, n = 3. The original expression has the value 432, and the answer 2m³n³ also has the value 432. ✓

Watch out. Subtract the indices (top minus bottom), and divide the numbers. Do not subtract the numbers.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. A number (or bracket) to the power zero is 1.

1. Remember the rule for a zero index. Any non-zero number raised to the power 0 equals 1. This is because aⁿ ÷ aⁿ = 1, and by the division law it is also aⁿ⁻ⁿ = a⁰.
   a⁰ = 1
2. Replace every power with index 0 by 1, and work out the rest. Only the part that has the 0 index becomes 1. Anything else stays.
   7⁰ = 1
3. Write the answer.
   1

Check. Look again: each 0 index was changed to 1 and nothing else was changed. ✓

Watch out. 5⁰ = 1, not 0 and not 5. And in 5x⁰ only the x has the index 0, so it is 5 × 1 = 5.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. A number (or bracket) to the power zero is 1.

1. Remember the rule for a zero index. Any non-zero number raised to the power 0 equals 1. This is because aⁿ ÷ aⁿ = 1, and by the division law it is also aⁿ⁻ⁿ = a⁰.
   a⁰ = 1
2. Replace every power with index 0 by 1, and work out the rest. Only the part that has the 0 index becomes 1. Anything else stays.
   100⁰ = 1
3. Write the answer.
   1

Check. Look again: each 0 index was changed to 1 and nothing else was changed. ✓

Watch out. 5⁰ = 1, not 0 and not 5. And in 5x⁰ only the x has the index 0, so it is 5 × 1 = 5.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. A number (or bracket) to the power zero is 1.

1. Remember the rule for a zero index. Any non-zero number raised to the power 0 equals 1. This is because aⁿ ÷ aⁿ = 1, and by the division law it is also aⁿ⁻ⁿ = a⁰.
   a⁰ = 1
2. Replace every power with index 0 by 1, and work out the rest. The bracket (−3) is a non-zero number, so its zero power is 1.
   (−3)⁰ = 1
3. Write the answer.
   1

Check. Look again: each 0 index was changed to 1 and nothing else was changed. ✓

Watch out. 5⁰ = 1, not 0 and not 5. And in 5x⁰ only the x has the index 0, so it is 5 × 1 = 5.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. A number (or bracket) to the power zero is 1.

1. Remember the rule for a zero index. Any non-zero number raised to the power 0 equals 1. This is because aⁿ ÷ aⁿ = 1, and by the division law it is also aⁿ⁻ⁿ = a⁰.
   a⁰ = 1
2. Replace every power with index 0 by 1, and work out the rest. Only the part that has the 0 index becomes 1. Anything else stays.
   2⁰ = 1 and 3⁰ = 1
   1 + 1 = 2
3. Write the answer.
   2

Check. Look again: each 0 index was changed to 1 and nothing else was changed. ✓

Watch out. 5⁰ = 1, not 0 and not 5. And in 5x⁰ only the x has the index 0, so it is 5 × 1 = 5.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. A number (or bracket) to the power zero is 1.

1. Remember the rule for a zero index. Any non-zero number raised to the power 0 equals 1. This is because aⁿ ÷ aⁿ = 1, and by the division law it is also aⁿ⁻ⁿ = a⁰.
   a⁰ = 1
2. Replace every power with index 0 by 1, and work out the rest. Only the part that has the 0 index becomes 1. Anything else stays.
   4⁰ = 1
   5 × 1 = 5
3. Write the answer.
   5

Check. Look again: each 0 index was changed to 1 and nothing else was changed. ✓

Watch out. 5⁰ = 1, not 0 and not 5. And in 5x⁰ only the x has the index 0, so it is 5 × 1 = 5.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ. For a fraction, flip it upside down and make the index positive.

1. Spot the negative index. A negative index does not make the answer negative. It means "one over".
   2⁻¹: the index is −1
2. Turn it into a reciprocal with a positive index. The rule is a⁻ⁿ = 1aⁿ. (Check with a pattern: 2² = 4, 2¹ = 2, 2⁰ = 1, and each step is ÷ 2, so 2⁻¹ = 12.)
   2⁻¹ = 12¹
3. Work out the bottom. 2¹ means 1 lots of 2 multiplied together.
   2¹ = 2 = 2
4. Write the answer as a fraction.
   2⁻¹ = 12

Check. Multiply the answer by 2¹: 12 × 2 = 1, and 2⁻¹ × 2¹ = 2⁰ = 1. ✓

Watch out. A negative index does not give a negative answer: 2⁻¹ = 12, not a negative number.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ. For a fraction, flip it upside down and make the index positive.

1. Spot the negative index. A negative index does not make the answer negative. It means "one over".
   3⁻²: the index is −2
2. Turn it into a reciprocal with a positive index. The rule is a⁻ⁿ = 1aⁿ. (Check with a pattern: 3² = 9, 3¹ = 3, 3⁰ = 1, and each step is ÷ 3, so 3⁻¹ = 13.)
   3⁻² = 13²
3. Work out the bottom. 3² means 2 lots of 3 multiplied together.
   3² = 3 × 3 = 9
4. Write the answer as a fraction.
   3⁻² = 19

Check. Multiply the answer by 3²: 19 × 9 = 1, and 3⁻² × 3² = 3⁰ = 1. ✓

Watch out. A negative index does not give a negative answer: 3⁻² = 19, not a negative number.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ. For a fraction, flip it upside down and make the index positive.

1. Spot the negative index. A negative index does not make the answer negative. It means "one over".
   10⁻³: the index is −3
2. Turn it into a reciprocal with a positive index. The rule is a⁻ⁿ = 1aⁿ. (Check with a pattern: 10² = 100, 10¹ = 10, 10⁰ = 1, and each step is ÷ 10, so 10⁻¹ = 110.)
   10⁻³ = 110³
3. Work out the bottom. 10³ means 3 lots of 10 multiplied together.
   10³ = 10 × 10 × 10 = 1000
4. Write the answer as a fraction.
   10⁻³ = 11000

Check. Multiply the answer by 10³: 11000 × 1000 = 1, and 10⁻³ × 10³ = 10⁰ = 1. ✓

Watch out. A negative index does not give a negative answer: 10⁻³ = 11000, not a negative number.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ. For a fraction, flip it upside down and make the index positive.

1. Spot the negative index. A negative index does not make the answer negative. It means "one over".
   5⁻²: the index is −2
2. Turn it into a reciprocal with a positive index. The rule is a⁻ⁿ = 1aⁿ. (Check with a pattern: 5² = 25, 5¹ = 5, 5⁰ = 1, and each step is ÷ 5, so 5⁻¹ = 15.)
   5⁻² = 15²
3. Work out the bottom. 5² means 2 lots of 5 multiplied together.
   5² = 5 × 5 = 25
4. Write the answer as a fraction.
   5⁻² = 125

Check. Multiply the answer by 5²: 125 × 25 = 1, and 5⁻² × 5² = 5⁰ = 1. ✓

Watch out. A negative index does not give a negative answer: 5⁻² = 125, not a negative number.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ. For a fraction, flip it upside down and make the index positive.

1. Spot the negative index. A negative index does not make the answer negative. It means "one over".
   4⁻¹: the index is −1
2. Turn it into a reciprocal with a positive index. The rule is a⁻ⁿ = 1aⁿ. (Check with a pattern: 4² = 16, 4¹ = 4, 4⁰ = 1, and each step is ÷ 4, so 4⁻¹ = 14.)
   4⁻¹ = 14¹
3. Work out the bottom. 4¹ means 1 lots of 4 multiplied together.
   4¹ = 4 = 4
4. Write the answer as a fraction.
   4⁻¹ = 14

Check. Multiply the answer by 4¹: 14 × 4 = 1, and 4⁻¹ × 4¹ = 4⁰ = 1. ✓

Watch out. A negative index does not give a negative answer: 4⁻¹ = 14, not a negative number.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ. For a fraction, flip it upside down and make the index positive.

1. Spot the negative index. A negative index does not make the answer negative. It means "one over".
   2⁻⁴: the index is −4
2. Turn it into a reciprocal with a positive index. The rule is a⁻ⁿ = 1aⁿ. (Check with a pattern: 2² = 4, 2¹ = 2, 2⁰ = 1, and each step is ÷ 2, so 2⁻¹ = 12.)
   2⁻⁴ = 12⁴
3. Work out the bottom. 2⁴ means 4 lots of 2 multiplied together.
   2⁴ = 2 × 2 × 2 × 2 = 16
4. Write the answer as a fraction.
   2⁻⁴ = 116

Check. Multiply the answer by 2⁴: 116 × 16 = 1, and 2⁻⁴ × 2⁴ = 2⁰ = 1. ✓

Watch out. A negative index does not give a negative answer: 2⁻⁴ = 116, not a negative number.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the bottom of the fraction as a power, then use a⁻ⁿ = 1/aⁿ backwards.

1. Look at the fraction. It has 1 on the top and a number on the bottom, so it is "one over" something. A negative index means "one over".
   18
2. Write the denominator as a power of the base. 8 = 2 × 2 × 2 = 2³
   18 = 12³
3. Use the rule for a negative index. "One over a power" is the same as the power with a negative index: 1aⁿ = a⁻ⁿ.
   12³ = 2⁻³

Check. Convert back: 2⁻³ = 12³ = 18. ✓

Watch out. The number on the bottom becomes the base. The index gets the minus sign.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the bottom of the fraction as a power, then use a⁻ⁿ = 1/aⁿ backwards.

1. Look at the fraction. It has 1 on the top and a number on the bottom, so it is "one over" something. A negative index means "one over".
   19
2. Write the denominator as a power of the base. 9 = 3 × 3 = 3²
   19 = 13²
3. Use the rule for a negative index. "One over a power" is the same as the power with a negative index: 1aⁿ = a⁻ⁿ.
   13² = 3⁻²

Check. Convert back: 3⁻² = 13² = 19. ✓

Watch out. The number on the bottom becomes the base. The index gets the minus sign.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the bottom of the fraction as a power, then use a⁻ⁿ = 1/aⁿ backwards.

1. Look at the fraction. It has 1 on the top and a number on the bottom, so it is "one over" something. A negative index means "one over".
   1100
2. Write the denominator as a power of the base. 100 = 10 × 10 = 10²
   1100 = 110²
3. Use the rule for a negative index. "One over a power" is the same as the power with a negative index: 1aⁿ = a⁻ⁿ.
   110² = 10⁻²

Check. Convert back: 10⁻² = 110² = 1100. ✓

Watch out. The number on the bottom becomes the base. The index gets the minus sign.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the bottom of the fraction as a power, then use a⁻ⁿ = 1/aⁿ backwards.

1. Look at the fraction. It has 1 on the top and a number on the bottom, so it is "one over" something. A negative index means "one over".
   1125
2. Write the denominator as a power of the base. 125 = 5 × 5 × 5 = 5³
   1125 = 15³
3. Use the rule for a negative index. "One over a power" is the same as the power with a negative index: 1aⁿ = a⁻ⁿ.
   15³ = 5⁻³

Check. Convert back: 5⁻³ = 15³ = 1125. ✓

Watch out. The number on the bottom becomes the base. The index gets the minus sign.

Key words. In 3⁴, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 3⁴ = 3 × 3 × 3 × 3 = 81.

Method. Write the bottom of the fraction as a power, then use a⁻ⁿ = 1/aⁿ backwards.

1. Look at the fraction. It has 1 on the top and a number on the bottom, so it is "one over" something. A negative index means "one over".
   132
2. Write the denominator as a power of the base. 32 = 2 × 2 × 2 × 2 × 2 = 2⁵
   132 = 12⁵
3. Use the rule for a negative index. "One over a power" is the same as the power with a negative index: 1aⁿ = a⁻ⁿ.
   12⁵ = 2⁻⁵