GCSE Maths Standard Form, Indices and Surds: Zero and negative indices (Foundation)

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

Check. Convert back: 2⁻⁵ = 12⁵ = 132. ✓

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. 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 means "reciprocal". The reciprocal of a fraction is the fraction turned upside down.
   (12)⁻³: the index is −3
2. Flip the fraction and change the index to positive. Negative index = reciprocal. The reciprocal of 12 is 21.
   (12)⁻³ = (21)³
3. Raise the top and the bottom to the power. The index 3 applies to the numerator and to the denominator.
   2³1³ = 81
4. Write the answer (simplify if possible).
   8

Check. Multiply the answer by (12)³ = 18: the product is 1, as (12)⁻³ × (12)³ = (12)⁰ = 1. ✓

Watch out. A negative index does not give a negative answer: (12)⁻³ = 8, 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 means "reciprocal". The reciprocal of a fraction is the fraction turned upside down.
   (23)⁻²: the index is −2
2. Flip the fraction and change the index to positive. Negative index = reciprocal. The reciprocal of 23 is 32.
   (23)⁻² = (32)²
3. Raise the top and the bottom to the power. The index 2 applies to the numerator and to the denominator.
   3²2² = 94
4. Write the answer (simplify if possible).
   94

Check. Multiply the answer by (23)² = 49: the product is 1, as (23)⁻² × (23)² = (23)⁰ = 1. ✓

Watch out. A negative index does not give a negative answer: (23)⁻² = 94, 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 means "reciprocal". The reciprocal of a fraction is the fraction turned upside down.
   (15)⁻²: the index is −2
2. Flip the fraction and change the index to positive. Negative index = reciprocal. The reciprocal of 15 is 51.
   (15)⁻² = (51)²
3. Raise the top and the bottom to the power. The index 2 applies to the numerator and to the denominator.
   5²1² = 251
4. Write the answer (simplify if possible).
   25

Check. Multiply the answer by (15)² = 125: the product is 1, as (15)⁻² × (15)² = (15)⁰ = 1. ✓

Watch out. A negative index does not give a negative answer: (15)⁻² = 25, 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 means "reciprocal". The reciprocal of a fraction is the fraction turned upside down.
   (34)⁻²: the index is −2
2. Flip the fraction and change the index to positive. Negative index = reciprocal. The reciprocal of 34 is 43.
   (34)⁻² = (43)²
3. Raise the top and the bottom to the power. The index 2 applies to the numerator and to the denominator.
   4²3² = 169
4. Write the answer (simplify if possible).
   169

Check. Multiply the answer by (34)² = 916: the product is 1, as (34)⁻² × (34)² = (34)⁰ = 1. ✓

Watch out. A negative index does not give a negative answer: (34)⁻² = 169, 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 means "reciprocal". The reciprocal of a fraction is the fraction turned upside down.
   (25)⁻¹: the index is −1
2. Flip the fraction and change the index to positive. Negative index = reciprocal. The reciprocal of 25 is 52.
   (25)⁻¹ = (52)¹
3. Raise the top and the bottom to the power. The index 1 applies to the numerator and to the denominator.
   5¹2¹ = 52
4. Write the answer (simplify if possible).
   52

Check. Multiply the answer by (25)¹ = 25: the product is 1, as (25)⁻¹ × (25)¹ = (25)⁰ = 1. ✓

Watch out. A negative index does not give a negative answer: (25)⁻¹ = 52, 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. 10 to a negative power is one over a power of 10: it is a small decimal.

1. Use the rule for a negative index. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ.
   10⁻¹ = 110¹
2. Work out the bottom. 10¹ is 1 followed by 1 zeros.
   10¹ = 10, so 10⁻¹ = 110
3. Write it as a decimal. Dividing by 10 moves the decimal point one place to the left, 1 times, starting from 1.
   1 → 0.1

Check. 1 ÷ 10 = 0.1. ✓

Watch out. 10⁻² is 0.01, not −100 and not −20.

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. 10 to a negative power is one over a power of 10: it is a small decimal.

1. Use the rule for a negative index. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ.
   10⁻² = 110²
2. Work out the bottom. 10² is 1 followed by 2 zeros.
   10² = 100, so 10⁻² = 1100
3. Write it as a decimal. Dividing by 10 moves the decimal point one place to the left, 2 times, starting from 1.
   1 → 0.1 → 0.01

Check. 1 ÷ 100 = 0.01. ✓

Watch out. 10⁻² is 0.01, not −100 and not −20.

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. 10 to a negative power is one over a power of 10: it is a small decimal.

1. Use the rule for a negative index. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ.
   10⁻³ = 110³
2. Work out the bottom. 10³ is 1 followed by 3 zeros.
   10³ = 1000, so 10⁻³ = 11000
3. Write it as a decimal. Dividing by 10 moves the decimal point one place to the left, 3 times, starting from 1.
   1 → 0.1 → 0.01 → 0.001

Check. 1 ÷ 1000 = 0.001. ✓

Watch out. 10⁻² is 0.01, not −100 and not −20.

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. 10 to a negative power is one over a power of 10: it is a small decimal.

1. Use the rule for a negative index. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ.
   10⁻⁴ = 110⁴
2. Work out the bottom. 10⁴ is 1 followed by 4 zeros.
   10⁴ = 10,000, so 10⁻⁴ = 110,000
3. Write it as a decimal. Dividing by 10 moves the decimal point one place to the left, 4 times, starting from 1.
   1 → 0.1 → 0.01 → 0.001 → 0.0001

Check. 1 ÷ 10,000 = 0.0001. ✓

Watch out. 10⁻² is 0.01, not −100 and not −20.

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. 10 to a negative power is one over a power of 10: it is a small decimal.

1. Use the rule for a negative index. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ.
   10⁻⁵ = 110⁵
2. Work out the bottom. 10⁵ is 1 followed by 5 zeros.
   10⁵ = 100,000, so 10⁻⁵ = 1100,000
3. Write it as a decimal. Dividing by 10 moves the decimal point one place to the left, 5 times, starting from 1.
   1 → 0.1 → 0.01 → 0.001 → 0.0001 → 0.00001

Check. 1 ÷ 100,000 = 0.00001. ✓

Watch out. 10⁻² is 0.01, not −100 and not −20.

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.
   10⁰ = 1
   Pattern: 1000, 100, 10, then 10 ÷ 10 = 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. 10 to a negative power is one over a power of 10: it is a small decimal.

1. Use the rule for a negative index. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ.
   10⁻¹ = 110¹
2. Work out the bottom. 10¹ is 1 followed by 1 zeros.
   10¹ = 10, so 10⁻¹ = 110
3. Write it as a decimal. Dividing by 10 moves the decimal point one place to the left, 1 times, starting from 1.
   1 → 0.1

Check. 1 ÷ 10 = 0.1. ✓

Watch out. 10⁻² is 0.01, not −100 and not −20.

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. 10 to a negative power is one over a power of 10: it is a small decimal.

1. Use the rule for a negative index. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ.
   10⁻² = 110²
2. Work out the bottom. 10² is 1 followed by 2 zeros.
   10² = 100, so 10⁻² = 1100
3. Write it as a decimal. Dividing by 10 moves the decimal point one place to the left, 2 times, starting from 1.
   1 → 0.1 → 0.01

Check. 1 ÷ 100 = 0.01. ✓

Watch out. 10⁻² is 0.01, not −100 and not −20.

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. Say what the rule means. The index counts the lots of the base. Multiplying powers joins the lots together.
   5² has 2 lots of 5; 5⁻⁴ has −4 lots of 5.
3. Add the indices. Joining the lots means adding the number of lots. The rule works for negative indices too, so take care with the signs.
   2 + (−4) = −2
4. Write the answer as a single power. Keep the same base and use the new index.
   5² × 5⁻⁴ = 5⁻²
5. Work out the value. A negative index means "one over" the positive power (see section 6.2).
   5⁻² = 15² = 125

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 2.
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.
   2³ has 3 lots of 2; we take away 5 lots.
3. Subtract the indices. The number of lots left is the first index take away the second.
   3 − 5 = −2
4. Write the answer as a single power. Keep the same base and use the new index.
   2³ ÷ 2⁵ = 2⁻²
5. Work out the value. A negative index means "one over" the positive power (see section 6.2).
   2⁻² = 12² = 14

Check. Work out the numbers: 2³ = 8 and 2⁵ = 32. 8 ÷ 32 = 14, and 2⁻² = 14. ✓

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. Say what the rule means. The outer index −1 says how many lots of 3² to multiply. Each lot contributes 2 lots of 3.
   2 × (−1) = −2
2. Multiply the indices. Each of the outer lots holds m of the base, so the total is m × n.
   2 × (−1) = −2
3. Write the answer as a single power. Keep the same base and use the new index.
   (3²)⁻¹ = 3⁻²
4. Work out the value. A negative index means "one over" the positive power (see section 6.2).
   3⁻² = 13² = 19

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. Say what the rule means. The index counts the lots of the base. Multiplying powers joins the lots together.
   10⁻³ has −3 lots of 10; 10⁵ has 5 lots of 10.
3. Add the indices. Joining the lots means adding the number of lots. The rule works for negative indices too, so take care with the signs.
   −3 + 5 = 2
4. Write the answer as a single power. Keep the same base and use the new index.
   10⁻³ × 10⁵ = 10²
5. Work out the value. 10² means 2 lots of 10 multiplied together.
   10² = 10 × 10 = 100

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. Say what the rule means. The index counts the lots of the base. Multiplying powers joins the lots together.
   7⁻² has −2 lots of 7; 7⁻¹ has −1 lots of 7.
3. Add the indices. Joining the lots means adding the number of lots. The rule works for negative indices too, so take care with the signs.
   −2 + (−1) = −3
4. Write the answer as a single power. Keep the same base and use the new index.
   7⁻² × 7⁻¹ = 7⁻³
5. Work out the value. A negative index means "one over" the positive power (see section 6.2).
   7⁻³ = 17³ = 1343

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. 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.)
   x² ÷ x⁶ = x^{2 − 6} = x⁻⁴ = 1x⁴
2. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   x⁻⁴
3. Write with positive indices. A negative index means "one over": a⁻ⁿ = 1/aⁿ.
   x⁻⁴ = 1x⁴

Check. Choose numbers: x = 2. The original expression has the value 116, and the answer x⁻⁴ also has the value 116. ✓

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.)
   y³ ÷ y⁸ = y^{3 − 8} = y⁻⁵ = 1y⁵
2. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   y⁻⁵
3. Write with positive indices. A negative index means "one over": a⁻ⁿ = 1/aⁿ.
   y⁻⁵ = 1y⁵

Check. Choose numbers: y = 3. The original expression has the value 1243, and the answer y⁻⁵ also has the value 1243. ✓

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. 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.)
   a² × a⁻⁵ = a^{2 + (−5)} = a⁻³
2. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   a⁻³
3. Write with positive indices. A negative index means "one over": a⁻ⁿ = 1/aⁿ.
   a⁻³ = 1a³

Check. Choose numbers: a = 2. The original expression has the value 18, and the answer a⁻³ also has the value 18. ✓

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.
   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.)
   x⁻² × x⁵ = x^{−2 + 5} = x³
3. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   12x³

Check. Choose numbers: x = 2. The original expression has the value 96, and the answer 12x³ also has the value 96. ✓

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.

Method. Write the number on the right as a power of the same base, then compare the indices.

1. Write the powers of the base in a list. We are looking for the power of 2 that equals 32. A list makes it easy to spot.
   2⁰ = 1
   2¹ = 2
   2² = 4
   2³ = 8
   2⁴ = 16
   2⁵ = 32
2. Find the one that matches. Read down the list until you see 32.
   2⁵ = 32
3. Write the value of the unknown. The unknown n is the index of the matching power.
   n = 5

Check. Put it back: 2⁵ = 32. ✓

Watch out. If the right-hand side is a fraction such as 1/8, the index is negative.

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 number on the right as a power of the same base, then compare the indices.

1. Write the powers of the base in a list. We are looking for the power of 3 that equals 81. A list makes it easy to spot.
   3⁰ = 1
   3¹ = 3
   3² = 9
   3³ = 27
   3⁴ = 81
2. Find the one that matches. Read down the list until you see 81.
   3⁴ = 81
3. Write the value of the unknown. The unknown n is the index of the matching power.
   n = 4

Check. Put it back: 3⁴ = 81. ✓

Watch out. If the right-hand side is a fraction such as 1/8, the index is negative.

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 number on the right as a power of the same base, then compare the indices.

1. Write the powers of the base in a list. We are looking for the power of 5 that equals 125. A list makes it easy to spot.
   5⁰ = 1
   5¹ = 5
   5² = 25
   5³ = 125
   5⁴ = 625
2. Find the one that matches. Read down the list until you see 125.
   5³ = 125
3. Write the value of the unknown. The unknown n is the index of the matching power.
   n = 3

Check. Put it back: 5³ = 125. ✓

Watch out. If the right-hand side is a fraction such as 1/8, the index is negative.

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 number on the right as a power of the same base, then compare the indices.

1. Write the powers of the base in a list. We are looking for the power of 10 that equals 1000. A list makes it easy to spot.
   10⁰ = 1
   10¹ = 10
   10² = 100
   10³ = 1000
   10⁴ = 10,000
2. Find the one that matches. Read down the list until you see 1000.
   10³ = 1000
3. Write the value of the unknown. The unknown n is the index of the matching power.
   n = 3

Check. Put it back: 10³ = 1000. ✓

Watch out. If the right-hand side is a fraction such as 1/8, the index is negative.

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 number on the right as a power of the same base, then compare the indices.

1. Write the powers of the base in a list. We are looking for the power of 2 that equals 128. A list makes it easy to spot.
   2⁰ = 1
   2¹ = 2
   2² = 4
   2³ = 8
   2⁴ = 16
   2⁵ = 32
   2⁶ = 64
   2⁷ = 128
2. Find the one that matches. Read down the list until you see 128.
   2⁷ = 128
3. Write the value of the unknown. The unknown n is the index of the matching power.
   n = 7

Check. Put it back: 2⁷ = 128. ✓

Watch out. If the right-hand side is a fraction such as 1/8, the index is negative.

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 number on the right as a power of the same base, then compare the indices.

1. Write the powers of the base in a list. We are looking for the power of 2 that equals 18. A list makes it easy to spot.
   2⁻³ = 18
   2⁻² = 14
   2⁻¹ = 12
   2⁰ = 1
   2¹ = 2
   2² = 4
   2³ = 8
   2⁴ = 16
2. Find the one that matches. Read down the list until you see 18.
   2⁻³ = 18
3. Write the value of the unknown. The unknown n is the index of the matching power.
   n = −3

Check. Put it back: 2⁻³ = 18. ✓

Watch out. If the right-hand side is a fraction such as 1/8, the index is negative.

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 number on the right as a power of the same base, then compare the indices.

1. Write the powers of the base in a list. We are looking for the power of 3 that equals 19. A list makes it easy to spot.
   3⁻³ = 127
   3⁻² = 19
   3⁻¹ = 13
   3⁰ = 1
   3¹ = 3
   3² = 9
   3³ = 27
   3⁴ = 81
2. Find the one that matches. Read down the list until you see 19.
   3⁻² = 19
3. Write the value of the unknown. The unknown n is the index of the matching power.
   n = −2

Check. Put it back: 3⁻² = 19. ✓

Watch out. If the right-hand side is a fraction such as 1/8, the index is negative.

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 number on the right as a power of the same base, then compare the indices.

1. Write the powers of the base in a list. We are looking for the power of 10 that equals 1100. A list makes it easy to spot.
   10⁻³ = 11000
   10⁻² = 1100
   10⁻¹ = 110
   10⁰ = 1
   10¹ = 10
   10² = 100
   10³ = 1000
   10⁴ = 10,000
2. Find the one that matches. Read down the list until you see 1100.
   10⁻² = 1100
3. Write the value of the unknown. The unknown n is the index of the matching power.
   n = −2

Check. Put it back: 10⁻² = 1100. ✓

Watch out. If the right-hand side is a fraction such as 1/8, the index is negative.

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 number on the right as a power of the same base, then compare the indices.

1. Write the powers of the base in a list. We are looking for the power of 5 that equals 15. A list makes it easy to spot.
   5⁻³ = 1125
   5⁻² = 125
   5⁻¹ = 15
   5⁰ = 1
   5¹ = 5
   5² = 25
   5³ = 125
   5⁴ = 625
2. Find the one that matches. Read down the list until you see 15.
   5⁻¹ = 15
3. Write the value of the unknown. The unknown n is the index of the matching power.
   n = −1

Check. Put it back: 5⁻¹ = 15. ✓

Watch out. If the right-hand side is a fraction such as 1/8, the index is negative.

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 number on the right as a power of the same base, then compare the indices.

1. Write the powers of the base in a list. We are looking for the power of 4 that equals 116. A list makes it easy to spot.
   4⁻³ = 164
   4⁻² = 116
   4⁻¹ = 14
   4⁰ = 1
   4¹ = 4
   4² = 16
   4³ = 64
   4⁴ = 256
2. Find the one that matches. Read down the list until you see 116.
   4⁻² = 116
3. Write the value of the unknown. The unknown n is the index of the matching power.
   n = −2

Check. Put it back: 4⁻² = 116. ✓

Watch out. If the right-hand side is a fraction such as 1/8, the index is negative.

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 equal bases: if the powers are equal, the indices are equal.

1. Both sides have the same base. When the bases are equal, the indices must be equal too.
   Base 2 on both sides.
2. Make the indices equal. If a^p = a^q then p = q.
   x + 3 = 8
3. Solve the simple equation. Subtract 3 from both sides.
   x = 8 − 3 = 5

Check. Put x = 5 back: the left-hand index is 5 + 3 = 8, so we get 2⁸, which matches the right-hand side. ✓

Watch out. Check by substituting back into the original equation.

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 equal bases: if the powers are equal, the indices are equal.

1. Both sides have the same base. When the bases are equal, the indices must be equal too.
   Base 3 on both sides.
2. Make the indices equal. If a^p = a^q then p = q.
   x − 1 = 4
3. Solve the simple equation. Add 1 from both sides.
   x = 4 + 1 = 5

Check. Put x = 5 back: the left-hand index is 5 − 1 = 4, so we get 3⁴, which matches the right-hand side. ✓

Watch out. Check by substituting back into the original equation.

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 equal bases: if the powers are equal, the indices are equal.

1. Both sides have the same base. When the bases are equal, the indices must be equal too.
   Base 5 on both sides.
2. Make the indices equal. If a^p = a^q then p = q.
   x + 2 = 6
3. Solve the simple equation. Subtract 2 from both sides.
   x = 6 − 2 = 4

Check. Put x = 4 back: the left-hand index is 4 + 2 = 6, so we get 5⁶, which matches the right-hand side. ✓

Watch out. Check by substituting back into the original equation.

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 equal bases: if the powers are equal, the indices are equal.

1. Both sides have the same base. When the bases are equal, the indices must be equal too.
   Base 10 on both sides.
2. Make the indices equal. If a^p = a^q then p = q.
   x − 2 = 3
3. Solve the simple equation. Add 2 from both sides.
   x = 3 + 2 = 5

Check. Put x = 5 back: the left-hand index is 5 − 2 = 3, so we get 10³, which matches the right-hand side. ✓

Watch out. Check by substituting back into the original equation.

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 each power as a fraction, then add.

1. Write each power as a fraction. A negative index means one over the positive power.
   2⁻¹ = 12 and 2⁻² = 14
2. Find a common denominator. The LCM of 2 and 4 is 4.
   12 = 24
3. Add the numerators.
   24 + 14 = 34

Check. As decimals: 0.5 + 0.25 = 0.75 = 3/4. ✓

Watch out. Convert each power to a fraction before adding.

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 each power as a fraction, then add the fractions. Fractions need a common denominator.

1. Write each power as a fraction. A negative index means one over the positive power.
   3⁻¹ = 13 and 3⁻² = 19
2. Find a common denominator. The denominators are 3 and 9. 9 is a multiple of 3, so use 9.
   13 = 39
3. Add the numerators and keep the denominator. The pieces are now the same size.
   39 + 19 = 49

Check. As decimals: 0.333… + 0.111… = 0.444… and 4/9 = 0.444…. ✓

Watch out. Convert each power to a fraction before adding. You cannot add the indices when adding powers.

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 each power as a number or fraction, then add.

1. Work out each power. 5⁰ = 1 (zero index) and 5⁻¹ = one over 5.
   5⁰ = 1 and 5⁻¹ = 15
2. Add. Write 1 as a fraction with denominator 5 so the pieces match.
   1 + 15 = 55 + 15 = 65
3. Write as a mixed number if you wish.
   65 = 115

Check. As decimals: 1 + 0.2 = 1.2 and 6/5 = 1.2. ✓

Watch out. 5⁰ is 1, not 0.

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 both numbers as powers of 2, then use the multiplication law.

1. Write 8 as a power of 2. 2 × 2 × 2 = 8.
   8 = 2³
2. Write 1/32 as a power of 2. 32 = 2⁵, and one over 2⁵ is 2⁻⁵.
   132 = 2⁻⁵
3. Multiply: add the indices.
   2³ × 2⁻⁵ = 2^{3 + (−5)} = 2⁻²

Check. 8 × 132 = 832 = 14 and 2⁻² = 14. ✓

Watch out. Convert to the same base first.