GCSE Maths: Powers and the laws of indices (Foundation)

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To multiply powers of the same base, add the indices: a^m × aⁿ = a^m+n.

1. Write each power out in full. 3⁴ is 4 copies of 3 multiplied, and 3² is 2 copies.
   3⁴ = 3 × 3 × 3 × 3
   3² = 3 × 3
2. Join them into one multiplication and count the copies of 3. Multiplying the two powers puts all the copies together: 4 + 2 copies.
   4 + 2 = 6 copies of 3
3. Write the answer as a single power. The number of copies is the new index. This is the rule: add the indices when you multiply powers of the same base.
   3⁴ × 3² = 3⁶

Check. Work out each power: 81 × 9 = 729, which is 3⁶. ✓

Watch out. Add the indices, do not multiply them: 3⁴ × 3² = 3⁶, not 3⁸. And the base stays the same.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To multiply powers of the same base, add the indices: a^m × aⁿ = a^m+n.

1. Write each power out in full. 5³ is 3 copies of 5 multiplied, and 5⁶ is 6 copies.
   5³ = 5 × 5 × 5
   5⁶ = 5 × 5 × 5 × 5 × 5 × 5
2. Join them into one multiplication and count the copies of 5. Multiplying the two powers puts all the copies together: 3 + 6 copies.
   3 + 6 = 9 copies of 5
3. Write the answer as a single power. The number of copies is the new index. This is the rule: add the indices when you multiply powers of the same base.
   5³ × 5⁶ = 5⁹

Check. Work out each power: 125 × 15,625 = 1,953,125, which is 5⁹. ✓

Watch out. Add the indices, do not multiply them: 5³ × 5⁶ = 5⁹, not 5¹⁸. And the base stays the same.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To multiply powers of the same base, add the indices: a^m × aⁿ = a^m+n.

1. Write each power out in full. 2⁷ is 7 copies of 2 multiplied, and 2⁵ is 5 copies.
   2⁷ = 2 × 2 × 2 × 2 × 2 × 2 × 2
   2⁵ = 2 × 2 × 2 × 2 × 2
2. Join them into one multiplication and count the copies of 2. Multiplying the two powers puts all the copies together: 7 + 5 copies.
   7 + 5 = 12 copies of 2
3. Write the answer as a single power. The number of copies is the new index. This is the rule: add the indices when you multiply powers of the same base.
   2⁷ × 2⁵ = 2¹²

Check. Work out each power: 128 × 32 = 4096, which is 2¹². ✓

Watch out. Add the indices, do not multiply them: 2⁷ × 2⁵ = 2¹², not 2³⁵. And the base stays the same.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To multiply powers of the same base, add the indices: a^m × aⁿ = a^m+n.

1. Write each power out in full. 7² is 2 copies of 7 multiplied, and 7⁹ is 9 copies.
   7² = 7 × 7
   7⁹ = 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
2. Join them into one multiplication and count the copies of 7. Multiplying the two powers puts all the copies together: 2 + 9 copies.
   2 + 9 = 11 copies of 7
3. Write the answer as a single power. The number of copies is the new index. This is the rule: add the indices when you multiply powers of the same base.
   7² × 7⁹ = 7¹¹

Check. Count the copies of 7 in the original: 2 + 9 = 11. ✓

Watch out. Add the indices, do not multiply them: 7² × 7⁹ = 7¹¹, not 7¹⁸. And the base stays the same.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To multiply powers of the same base, add the indices: a^m × aⁿ = a^m+n.

1. Write each power out in full. 10⁴ is 4 copies of 10 multiplied, and 10³ is 3 copies.
   10⁴ = 10 × 10 × 10 × 10
   10³ = 10 × 10 × 10
2. Join them into one multiplication and count the copies of 10. Multiplying the two powers puts all the copies together: 4 + 3 copies.
   4 + 3 = 7 copies of 10
3. Write the answer as a single power. The number of copies is the new index. This is the rule: add the indices when you multiply powers of the same base.
   10⁴ × 10³ = 10⁷

Check. Count the copies of 10 in the original: 4 + 3 = 7. ✓

Watch out. Add the indices, do not multiply them: 10⁴ × 10³ = 10⁷, not 10¹². And the base stays the same.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To multiply powers of the same base, add the indices: a^m × aⁿ = a^m+n.

1. Write each power out in full. 4² is 2 copies of 4 multiplied, and 4⁵ is 5 copies.
   4² = 4 × 4
   4⁵ = 4 × 4 × 4 × 4 × 4
2. Join them into one multiplication and count the copies of 4. Multiplying the two powers puts all the copies together: 2 + 5 copies.
   2 + 5 = 7 copies of 4
3. Write the answer as a single power. The number of copies is the new index. This is the rule: add the indices when you multiply powers of the same base.
   4² × 4⁵ = 4⁷

Check. Work out each power: 16 × 1024 = 16,384, which is 4⁷. ✓

Watch out. Add the indices, do not multiply them: 4² × 4⁵ = 4⁷, not 4¹⁰. And the base stays the same.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To divide powers of the same base, subtract the indices: a^m ÷ aⁿ = a^m−n.

1. Write the division as a fraction with each power written in full. A fraction line means divide, and it lets us cancel.
   2⁹ ÷ 2⁴ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 22 × 2 × 2 × 2
2. Cancel the copies of 2 that appear on the top and the bottom. Each 2 on the bottom cancels with one on the top, because 2 ÷ 2 = 1. We cancel 4 of them.
   9 − 4 = 5 copies of 2 left on the top
3. Write the answer as a single power. The copies left on the top are the new index. This is the rule: subtract the indices when you divide powers of the same base.
   2⁹ ÷ 2⁴ = 2⁵

Check. 512 ÷ 16 = 32. ✓

Watch out. Subtract the indices, do not divide them. The base stays 2.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To divide powers of the same base, subtract the indices: a^m ÷ aⁿ = a^m−n.

1. Write the division as a fraction with each power written in full. A fraction line means divide, and it lets us cancel.
   5⁸ ÷ 5⁵ = 5 × 5 × 5 × 5 × 5 × 5 × 5 × 55 × 5 × 5 × 5 × 5
2. Cancel the copies of 5 that appear on the top and the bottom. Each 5 on the bottom cancels with one on the top, because 5 ÷ 5 = 1. We cancel 5 of them.
   8 − 5 = 3 copies of 5 left on the top
3. Write the answer as a single power. The copies left on the top are the new index. This is the rule: subtract the indices when you divide powers of the same base.
   5⁸ ÷ 5⁵ = 5³

Check. 390,625 ÷ 3125 = 125. ✓

Watch out. Subtract the indices, do not divide them. The base stays 5.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To divide powers of the same base, subtract the indices: a^m ÷ aⁿ = a^m−n.

1. Write the division as a fraction with each power written in full. A fraction line means divide, and it lets us cancel.
   3⁷ ÷ 3² = 3 × 3 × 3 × 3 × 3 × 3 × 33 × 3
2. Cancel the copies of 3 that appear on the top and the bottom. Each 3 on the bottom cancels with one on the top, because 3 ÷ 3 = 1. We cancel 2 of them.
   7 − 2 = 5 copies of 3 left on the top
3. Write the answer as a single power. The copies left on the top are the new index. This is the rule: subtract the indices when you divide powers of the same base.
   3⁷ ÷ 3² = 3⁵

Check. 2187 ÷ 9 = 243. ✓

Watch out. Subtract the indices, do not divide them. The base stays 3.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To divide powers of the same base, subtract the indices: a^m ÷ aⁿ = a^m−n.

1. Write the division as a fraction with each power written in full. A fraction line means divide, and it lets us cancel.
   7⁶ ÷ 7³ = 7 × 7 × 7 × 7 × 7 × 77 × 7 × 7
2. Cancel the copies of 7 that appear on the top and the bottom. Each 7 on the bottom cancels with one on the top, because 7 ÷ 7 = 1. We cancel 3 of them.
   6 − 3 = 3 copies of 7 left on the top
3. Write the answer as a single power. The copies left on the top are the new index. This is the rule: subtract the indices when you divide powers of the same base.
   7⁶ ÷ 7³ = 7³

Check. 117,649 ÷ 343 = 343. ✓

Watch out. Subtract the indices, do not divide them. The base stays 7.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To divide powers of the same base, subtract the indices: a^m ÷ aⁿ = a^m−n.

1. Write the division as a fraction with each power written in full. A fraction line means divide, and it lets us cancel.
   10⁸ ÷ 10⁵ = 10 × 10 × 10 × 10 × 10 × 10 × 10 × 1010 × 10 × 10 × 10 × 10
2. Cancel the copies of 10 that appear on the top and the bottom. Each 10 on the bottom cancels with one on the top, because 10 ÷ 10 = 1. We cancel 5 of them.
   8 − 5 = 3 copies of 10 left on the top
3. Write the answer as a single power. The copies left on the top are the new index. This is the rule: subtract the indices when you divide powers of the same base.
   10⁸ ÷ 10⁵ = 10³

Check. 100,000,000 ÷ 100,000 = 1000. ✓

Watch out. Subtract the indices, do not divide them. The base stays 10.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To divide powers of the same base, subtract the indices: a^m ÷ aⁿ = a^m−n.

1. Write the division as a fraction with each power written in full. A fraction line means divide, and it lets us cancel.
   6⁵ ÷ 6² = 6 × 6 × 6 × 6 × 66 × 6
2. Cancel the copies of 6 that appear on the top and the bottom. Each 6 on the bottom cancels with one on the top, because 6 ÷ 6 = 1. We cancel 2 of them.
   5 − 2 = 3 copies of 6 left on the top
3. Write the answer as a single power. The copies left on the top are the new index. This is the rule: subtract the indices when you divide powers of the same base.
   6⁵ ÷ 6² = 6³

Check. 7776 ÷ 36 = 216. ✓

Watch out. Subtract the indices, do not divide them. The base stays 6.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

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

1. Write the outer power as repeated multiplication of the bracket. (3²)⁴ means 4 copies of 3² multiplied together.
   (3²)⁴ = 3² × 3² × 3² × 3²
2. Each bracket holds 2 copies of 3. Count the copies altogether. 4 brackets with 2 copies in each.
   4 × 2 = 8 copies of 3
3. Write the answer as a single power. This is the rule: multiply the indices when you raise a power to a power.
   (3²)⁴ = 3⁸

Check. 9⁴ = 6561 and 3⁸ = 6561. ✓

Watch out. (3²)⁴ multiplies the indices (2 × 4 = 8). Do not add them: 2 + 4 = 6 is the rule for multiplying two separate powers.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

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

1. Write the outer power as repeated multiplication of the bracket. (5³)² means 2 copies of 5³ multiplied together.
   (5³)² = 5³ × 5³
2. Each bracket holds 3 copies of 5. Count the copies altogether. 2 brackets with 3 copies in each.
   2 × 3 = 6 copies of 5
3. Write the answer as a single power. This is the rule: multiply the indices when you raise a power to a power.
   (5³)² = 5⁶

Check. 125² = 15,625 and 5⁶ = 15,625. ✓

Watch out. (5³)² multiplies the indices (3 × 2 = 6). Do not add them: 3 + 2 = 5 is the rule for multiplying two separate powers.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

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

1. Write the outer power as repeated multiplication of the bracket. (2⁴)³ means 3 copies of 2⁴ multiplied together.
   (2⁴)³ = 2⁴ × 2⁴ × 2⁴
2. Each bracket holds 4 copies of 2. Count the copies altogether. 3 brackets with 4 copies in each.
   3 × 4 = 12 copies of 2
3. Write the answer as a single power. This is the rule: multiply the indices when you raise a power to a power.
   (2⁴)³ = 2¹²

Check. 16³ = 4096 and 2¹² = 4096. ✓

Watch out. (2⁴)³ multiplies the indices (4 × 3 = 12). Do not add them: 4 + 3 = 7 is the rule for multiplying two separate powers.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

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

1. Write the outer power as repeated multiplication of the bracket. (10²)⁵ means 5 copies of 10² multiplied together.
   (10²)⁵ = 10² × 10² × 10² × 10² × 10²
2. Each bracket holds 2 copies of 10. Count the copies altogether. 5 brackets with 2 copies in each.
   5 × 2 = 10 copies of 10
3. Write the answer as a single power. This is the rule: multiply the indices when you raise a power to a power.
   (10²)⁵ = 10¹⁰

Check. 5 copies of 2 copies is 10 copies. ✓

Watch out. (10²)⁵ multiplies the indices (2 × 5 = 10). Do not add them: 2 + 5 = 7 is the rule for multiplying two separate powers.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

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

1. Write the outer power as repeated multiplication of the bracket. (7³)² means 2 copies of 7³ multiplied together.
   (7³)² = 7³ × 7³
2. Each bracket holds 3 copies of 7. Count the copies altogether. 2 brackets with 3 copies in each.
   2 × 3 = 6 copies of 7
3. Write the answer as a single power. This is the rule: multiply the indices when you raise a power to a power.
   (7³)² = 7⁶

Check. 343² = 117,649 and 7⁶ = 117,649. ✓

Watch out. (7³)² multiplies the indices (3 × 2 = 6). Do not add them: 3 + 2 = 5 is the rule for multiplying two separate powers.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

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

1. Write the outer power as repeated multiplication of the bracket. (4²)³ means 3 copies of 4² multiplied together.
   (4²)³ = 4² × 4² × 4²
2. Each bracket holds 2 copies of 4. Count the copies altogether. 3 brackets with 2 copies in each.
   3 × 2 = 6 copies of 4
3. Write the answer as a single power. This is the rule: multiply the indices when you raise a power to a power.
   (4²)³ = 4⁶

Check. 16³ = 4096 and 4⁶ = 4096. ✓

Watch out. (4²)³ multiplies the indices (2 × 3 = 6). Do not add them: 2 + 3 = 5 is the rule for multiplying two separate powers.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To multiply powers of the same base, add the indices: a^m × aⁿ = a^m+n.

1. Write each power out in full. 2³ is 3 copies of 2 multiplied, and 2² is 2 copies.
   2³ = 2 × 2 × 2
   2² = 2 × 2
2. Join them into one multiplication and count the copies of 2. Multiplying the two powers puts all the copies together: 3 + 2 copies.
   3 + 2 = 5 copies of 2
3. Write the answer as a single power. The number of copies is the new index. This is the rule: add the indices when you multiply powers of the same base.
   2³ × 2² = 2⁵
4. Work out the value. Multiply the copies out.
   2⁵ = 32

Check. Work out each power: 8 × 4 = 32, which is 2⁵. ✓

Watch out. Add the indices, do not multiply them: 2³ × 2² = 2⁵, not 2⁶. And the base stays the same.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To divide powers of the same base, subtract the indices: a^m ÷ aⁿ = a^m−n.

1. Write the division as a fraction with each power written in full. A fraction line means divide, and it lets us cancel.
   3⁵ ÷ 3³ = 3 × 3 × 3 × 3 × 33 × 3 × 3
2. Cancel the copies of 3 that appear on the top and the bottom. Each 3 on the bottom cancels with one on the top, because 3 ÷ 3 = 1. We cancel 3 of them.
   5 − 3 = 2 copies of 3 left on the top
3. Write the answer as a single power. The copies left on the top are the new index. This is the rule: subtract the indices when you divide powers of the same base.
   3⁵ ÷ 3³ = 3²
4. Work out the value. Multiply the copies out.
   3² = 9

Check. 243 ÷ 27 = 9. ✓

Watch out. Subtract the indices, do not divide them. The base stays 3.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

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

1. Write the outer power as repeated multiplication of the bracket. (2²)³ means 3 copies of 2² multiplied together.
   (2²)³ = 2² × 2² × 2²
2. Each bracket holds 2 copies of 2. Count the copies altogether. 3 brackets with 2 copies in each.
   3 × 2 = 6 copies of 2
3. Write the answer as a single power. This is the rule: multiply the indices when you raise a power to a power.
   (2²)³ = 2⁶
4. Work out the value. Multiply the copies out.
   2⁶ = 64

Check. 4³ = 64 and 2⁶ = 64. ✓

Watch out. (2²)³ multiplies the indices (2 × 3 = 6). Do not add them: 2 + 3 = 5 is the rule for multiplying two separate powers.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To multiply powers of the same base, add the indices: a^m × aⁿ = a^m+n.

1. Write each power out in full. 10² is 2 copies of 10 multiplied, and 10³ is 3 copies.
   10² = 10 × 10
   10³ = 10 × 10 × 10
2. Join them into one multiplication and count the copies of 10. Multiplying the two powers puts all the copies together: 2 + 3 copies.
   2 + 3 = 5 copies of 10
3. Write the answer as a single power. The number of copies is the new index. This is the rule: add the indices when you multiply powers of the same base.
   10² × 10³ = 10⁵
4. Work out the value. Multiply the copies out.
   10⁵ = 100,000

Check. Work out each power: 100 × 1000 = 100,000, which is 10⁵. ✓

Watch out. Add the indices, do not multiply them: 10² × 10³ = 10⁵, not 10⁶. And the base stays the same.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To divide powers of the same base, subtract the indices: a^m ÷ aⁿ = a^m−n.

1. Write the division as a fraction with each power written in full. A fraction line means divide, and it lets us cancel.
   5⁴ ÷ 5² = 5 × 5 × 5 × 55 × 5
2. Cancel the copies of 5 that appear on the top and the bottom. Each 5 on the bottom cancels with one on the top, because 5 ÷ 5 = 1. We cancel 2 of them.
   4 − 2 = 2 copies of 5 left on the top
3. Write the answer as a single power. The copies left on the top are the new index. This is the rule: subtract the indices when you divide powers of the same base.
   5⁴ ÷ 5² = 5²
4. Work out the value. Multiply the copies out.
   5² = 25

Check. 625 ÷ 25 = 25. ✓

Watch out. Subtract the indices, do not divide them. The base stays 5.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

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

1. Write the outer power as repeated multiplication of the bracket. (3²)² means 2 copies of 3² multiplied together.
   (3²)² = 3² × 3²
2. Each bracket holds 2 copies of 3. Count the copies altogether. 2 brackets with 2 copies in each.
   2 × 2 = 4 copies of 3
3. Write the answer as a single power. This is the rule: multiply the indices when you raise a power to a power.
   (3²)² = 3⁴
4. Work out the value. Multiply the copies out.
   3⁴ = 81

Check. 9² = 81 and 3⁴ = 81. ✓

Watch out. (3²)² multiplies the indices (2 × 2 = 4). Do not add them: 2 + 2 = 4 is the rule for multiplying two separate powers.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the index law to write the left side as a single power, then make the indices equal.

1. Write the equation. We want the value of n.
   2ⁿ × 2³ = 2⁸
2. Use the multiplication law on the left. Add the indices.
   2ⁿ × 2³ = 2^{n + 3}
3. Make the indices equal. The bases are the same, so the indices must match.
   n + 3 = 8
4. Solve. Subtract 3 from both sides.
   n = 8 − 3 = 5

Check. 2⁵ × 2³ = 2⁸ = 2⁸. ✓

Watch out. Use the law in the correct direction: add for multiply, subtract for divide, multiply for a power of a power.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the index law to write the left side as a single power, then make the indices equal.

1. Write the equation. We want the value of n.
   7ⁿ × 7⁵ = 7⁹
2. Use the multiplication law on the left. Add the indices.
   7ⁿ × 7⁵ = 7^{n + 5}
3. Make the indices equal. The bases are the same, so the indices must match.
   n + 5 = 9
4. Solve. Subtract 5 from both sides.
   n = 9 − 5 = 4

Check. 7⁴ × 7⁵ = 7⁹ = 7⁹. ✓

Watch out. Use the law in the correct direction: add for multiply, subtract for divide, multiply for a power of a power.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the index law to write the left side as a single power, then make the indices equal.

1. Write the equation. We want the value of n.
   3ⁿ ÷ 3² = 3⁵
2. Use the division law on the left. Subtract the indices.
   3ⁿ ÷ 3² = 3^{n − 2}
3. Make the indices equal. The bases are the same, so the indices must match.
   n − 2 = 5
4. Solve. Add 2 to both sides.
   n = 5 + 2 = 7

Check. 3⁷ ÷ 3² = 3⁵ = 3⁵. ✓

Watch out. Use the law in the correct direction: add for multiply, subtract for divide, multiply for a power of a power.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the index law to write the left side as a single power, then make the indices equal.

1. Write the equation. We want the value of n.
   5ⁿ ÷ 5⁴ = 5³
2. Use the division law on the left. Subtract the indices.
   5ⁿ ÷ 5⁴ = 5^{n − 4}
3. Make the indices equal. The bases are the same, so the indices must match.
   n − 4 = 3
4. Solve. Add 4 to both sides.
   n = 3 + 4 = 7

Check. 5⁷ ÷ 5⁴ = 5³ = 5³. ✓

Watch out. Use the law in the correct direction: add for multiply, subtract for divide, multiply for a power of a power.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the index law to write the left side as a single power, then make the indices equal.

1. Write the equation. We want the value of n.
   (2ⁿ)³ = 2¹²
2. Use the power-of-a-power law on the left. Multiply the indices.
   (2ⁿ)³ = 2³ⁿ
3. Make the indices equal. The bases are the same, so the indices must match.
   3n = 12
4. Solve. Divide both sides by 3.
   n = 12 ÷ 3 = 4

Check. (2⁴)³ = 2¹² = 2¹². ✓

Watch out. Use the law in the correct direction: add for multiply, subtract for divide, multiply for a power of a power.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the index law to write the left side as a single power, then make the indices equal.

1. Write the equation. We want the value of n.
   (10ⁿ)² = 10⁶
2. Use the power-of-a-power law on the left. Multiply the indices.
   (10ⁿ)² = 10²ⁿ
3. Make the indices equal. The bases are the same, so the indices must match.
   2n = 6
4. Solve. Divide both sides by 2.
   n = 6 ÷ 2 = 3

Check. (10³)² = 10⁶ = 10⁶. ✓

Watch out. Use the law in the correct direction: add for multiply, subtract for divide, multiply for a power of a power.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the index laws one at a time, in BIDMAS order: add indices to multiply, subtract to divide, multiply for a power of a power.

1. Write the expression. Look for brackets and decide the order.
   (2⁵ × 2⁴) ÷ 2³
2. Do the multiplication on the top first. Multiplying powers of the same base: add the indices.
   2⁵ × 2⁴ = 2⁹   (5 + 4 = 9)
3. Now divide. Dividing powers of the same base: subtract the indices.
   2⁹ ÷ 2³ = 2⁶   (9 − 3 = 6)
4. Work out the value. Multiply the copies out.
   2⁶ = 64

Check. Evaluate the original with numbers: 32 × 16 ÷ 8 = 64. ✓

Watch out. Only combine indices when the bases are the same. 2³ × 3² cannot be written as one power.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the index laws one at a time, in BIDMAS order: add indices to multiply, subtract to divide, multiply for a power of a power.

1. Write the expression. Look for brackets and decide the order.
   (3²)³ × 3
2. Deal with the power of a power first. Raising a power to a power: multiply the indices.
   (3²)³ = 3⁶   (2 × 3 = 6)
3. Now multiply. Multiplying powers of the same base: add the indices.
   3⁶ × 3 = 3⁷   (6 + 1 = 7)
4. Work out the value. Multiply the copies out.
   3⁷ = 2187

Check. Evaluate the original with numbers: 729 × 3 = 2187. ✓

Watch out. Only combine indices when the bases are the same. 2³ × 3² cannot be written as one power.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the index laws one at a time, in BIDMAS order: add indices to multiply, subtract to divide, multiply for a power of a power.

1. Write the expression. Look for brackets and decide the order.
   2⁹ ÷ (2² × 2³)
2. Work out the bracket first. Multiplying powers of the same base: add the indices.
   2² × 2³ = 2⁵   (2 + 3 = 5)
3. Now divide. Dividing powers of the same base: subtract the indices.
   2⁹ ÷ 2⁵ = 2⁴   (9 − 5 = 4)
4. Work out the value. Multiply the copies out.
   2⁴ = 16

Check. Evaluate the original with numbers: 512 ÷ 32 = 16. ✓

Watch out. Only combine indices when the bases are the same. 2³ × 3² cannot be written as one power.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the index laws one at a time, in BIDMAS order: add indices to multiply, subtract to divide, multiply for a power of a power.

1. Write the expression. Look for brackets and decide the order.
   (5⁶ × 5³) ÷ 5⁴
2. Do the multiplication on the top first. Multiplying powers of the same base: add the indices.
   5⁶ × 5³ = 5⁹   (6 + 3 = 9)
3. Now divide. Dividing powers of the same base: subtract the indices.
   5⁹ ÷ 5⁴ = 5⁵   (9 − 4 = 5)
4. Work out the value. Multiply the copies out.
   5⁵ = 3125

Check. Evaluate the original with numbers: 15,625 × 125 ÷ 625 = 3125. ✓

Watch out. Only combine indices when the bases are the same. 2³ × 3² cannot be written as one power.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the index laws one at a time, in BIDMAS order: add indices to multiply, subtract to divide, multiply for a power of a power.

1. Write the expression. Look for brackets and decide the order.
   (2³)² × 2⁴
2. Deal with the power of a power first. Raising a power to a power: multiply the indices.
   (2³)² = 2⁶   (3 × 2 = 6)
3. Now multiply. Multiplying powers of the same base: add the indices.
   2⁶ × 2⁴ = 2¹⁰   (6 + 4 = 10)
4. Work out the value. Multiply the copies out.
   2¹⁰ = 1024

Check. Evaluate the original with numbers: 64 × 16 = 1024. ✓

Watch out. Only combine indices when the bases are the same. 2³ × 3² cannot be written as one power.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the index laws one at a time, in BIDMAS order: add indices to multiply, subtract to divide, multiply for a power of a power.

1. Write the expression. Look for brackets and decide the order.
   3¹⁰ ÷ (3³ × 3²)
2. Work out the bracket first. Multiplying powers of the same base: add the indices.
   3³ × 3² = 3⁵   (3 + 2 = 5)
3. Now divide. Dividing powers of the same base: subtract the indices.
   3¹⁰ ÷ 3⁵ = 3⁵   (10 − 5 = 5)
4. Work out the value. Multiply the copies out.
   3⁵ = 243

Check. Evaluate the original with numbers: 59,049 ÷ 243 = 243. ✓

Watch out. Only combine indices when the bases are the same. 2³ × 3² cannot be written as one power.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the number as a power of the base by multiplying up.

1. Write the equation. We need the power of 2 that gives 64.
   2^x = 64
2. Write the powers of 2 until you reach 64. The index is the number of times we multiplied by the base.
   2 = 2
   2² = 4
   2³ = 8
   2⁴ = 16
   2⁵ = 32
   2⁶ = 64
3. State the value of x. 64 is 2⁶.
   x = 6

Check. 2⁶ = 64. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the number as a power of the base by multiplying up.

1. Write the equation. We need the power of 3 that gives 81.
   3^x = 81
2. Write the powers of 3 until you reach 81. The index is the number of times we multiplied by the base.
   3 = 3
   3² = 9
   3³ = 27
   3⁴ = 81
3. State the value of x. 81 is 3⁴.
   x = 4

Check. 3⁴ = 81. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the number as a power of the base by multiplying up.

1. Write the equation. We need the power of 5 that gives 125.
   5^x = 125
2. Write the powers of 5 until you reach 125. The index is the number of times we multiplied by the base.
   5 = 5
   5² = 25
   5³ = 125
3. State the value of x. 125 is 5³.
   x = 3

Check. 5³ = 125. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the number as a power of the base by multiplying up.

1. Write the equation. We need the power of 2 that gives 1024.
   2^x = 1024
2. Write the powers of 2 until you reach 1024. The index is the number of times we multiplied by the base.
   2 = 2
   2² = 4
   2³ = 8
   2⁴ = 16
   2⁵ = 32
   2⁶ = 64
   2⁷ = 128
   2⁸ = 256
   2⁹ = 512
   2¹⁰ = 1024
3. State the value of x. 1024 is 2¹⁰.
   x = 10

Check. 2¹⁰ = 1024. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the number as a power of the base by multiplying up.

1. Write the equation. We need the power of 4 that gives 256.
   4^x = 256
2. Write the powers of 4 until you reach 256. The index is the number of times we multiplied by the base.
   4 = 4
   4² = 16
   4³ = 64
   4⁴ = 256
3. State the value of x. 256 is 4⁴.
   x = 4

Check. 4⁴ = 256. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the number as a power of the base by multiplying up.

1. Write the equation. We need the power of 3 that gives 729.
   3^x = 729
2. Write the powers of 3 until you reach 729. The index is the number of times we multiplied by the base.
   3 = 3
   3² = 9
   3³ = 27
   3⁴ = 81
   3⁵ = 243
   3⁶ = 729
3. State the value of x. 729 is 3⁶.
   x = 6

Check. 3⁶ = 729. ✓

Method. Each fold multiplies the thickness by 2. After 10 folds the thickness has been multiplied by 2 ten times.

1. Count the doublings. Ten folds mean ten lots of × 2.
   2 × 2 × ... (10 times) = 2¹⁰
2. Write the thickness. Multiply the starting thickness by this.
   0.1 × 2¹⁰ mm

Check. After 1 fold: 0.1 × 2¹ = 0.2 mm, which is double 0.1 mm. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Work out 2¹⁰ first, then multiply by 0.1.

1. Work out 2¹⁰. Use the powers of 2 you know: 2⁵ = 32 and 2¹⁰ = 32 × 32.
   32 × 32 = 1024
2. Multiply by 0.1. Multiplying by 0.1 moves each digit one place to the right.
   0.1 × 1024 = 102.4 mm

Check. 1024 ÷ 10 = 102.4. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Work it out by writing the powers in full.

1. Write out the multiplication. Count the copies of 2.
   2³ × 2⁴ = (2 × 2 × 2) × (2 × 2 × 2 × 2)
2. Count the 2s. There are 3 + 4 = 7 copies of 2.
   = 2⁷ = 128
3. Compare with Ali. 4⁷ = 16 384, which is far too big.
   Ali multiplied the bases (2 × 2 = 4). The base must stay as 2 and only the indices are added.

Check. 2³ × 2⁴ = 8 × 16 = 128 = 2⁷. ✓

Watch out. Never multiply the bases. The base stays the same.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the power of a power in full.

1. Write out the power of a power. (2³)² means two lots of 2³.
   (2³)² = 2³ × 2³ = 8 × 8
2. Count the copies of 2. Two groups of 3.
   3 + 3 = 6, so the answer is 2⁶ = 64
3. Compare with Ben. Ben added 3 + 2.
   2⁵ = 32 is not 64, so Ben is wrong. For a power of a power, multiply the indices: 3 × 2 = 6.

Check. 8 × 8 = 64 = 2⁶. ✓

Watch out. Add indices for two separate powers multiplied; multiply indices for a power of a power.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Work out both sides.

1. Work out the left side. Evaluate each power, then add.
   3² = 9 and 3³ = 27, so 9 + 27 = 36
2. Work out the right side.
   3⁵ = 243
3. Compare. The index laws are for multiplying and dividing, not adding.
   36 ≠ 243, so Cara is wrong.

Check. 3² × 3³ = 3⁵ = 243, which is the multiplication, not the addition. ✓

Watch out. Adding powers does not follow the index laws.