Year 2 · Unit 1 · Lesson 2 of 5

Index Laws: Add the Powers, Don't Multiply Them

About 15 minutes · index notation · multiplying and dividing powers · a power of a power · the zero index · index laws in algebra · no calculator · exit quiz

Hook

Your friend says 2⁵ × 2³ = 2¹⁵, because 5 × 3 = 15.

Is your friend right? Write the powers out in full to check.

No! Written out, 2⁵ × 2³ is five 2s multiplied, then three more 2s: eight 2s altogether. So 2⁵ × 2³ = 2⁸ = 256. When you multiply powers of the same number, you add the indices.

PowerWritten out2s
2⁵2 × 2 × 2 × 2 × 25
2³2 × 2 × 23
2⁵ × 2³all of them8

2⁵ × 2³ has 5 + 3 = 8 twos, so it is 2⁸.

Learn

Master the essentials

LawRule · example
Multiplyam × an = am + n
3⁴ × 3⁵ = 3⁹
Divideam ÷ an = am − n
7⁸ ÷ 7³ = 7⁵
Power of a power(am)n = amn
(5²)⁴ = 5⁸
Index zeroa⁰ = 1
7⁰ = 1

The index laws only work when the base is the same.

Index notation

The index (power) says how many times the base is multiplied by itself.

5³ = 5 × 5 × 5 = 125

Multiplying

Same base: add the indices.

3⁴ × 3⁵ = 3⁹

Dividing

Same base: subtract the indices.

7⁸ ÷ 7³ = 7⁵

Power of a power

Multiply the indices.

(5²)⁴ = 5⁸

Index zero

Any number (except 0) to the power 0 is 1, because 4³ ÷ 4³ = 4⁰ = 1.

7⁰ = 1 · 100⁰ = 1

In algebra

The same laws work with letters.

x³ × x⁴ = x⁷ · y⁸ ÷ y² = y⁶

Worked examples

See it done

1. Simplify 3⁴ × 3⁵. Leave your answer as a power.

  1. Same base (3), multiplying: add the indices.
  2. 4 + 5 = 9.

3⁹

2. Simplify 7⁸ ÷ 7³.

  1. Same base (7), dividing: subtract the indices.
  2. 8 − 3 = 5.

7⁵

3. Simplify (x³)⁴ ÷ x⁵.

  1. Power of a power: multiply, 3 × 4 = 12, so x¹².
  2. Then divide: subtract, 12 − 5 = 7.

x⁷

Common mistake: Multiplying the indices when you should add them. 2⁵ × 2³ is 2⁸, not 2¹⁵. You only multiply indices for a power of a power, like (2⁵)³ = 2¹⁵. And never multiply the bases: 2⁵ × 2³ is not 4⁸.

PractiseNo calculator

10 questions with instant feedback

Choose the correct answer. Check the base is the same, then decide: add, subtract or multiply the indices? Questions 1–3 are warm-ups, 4–7 are at exam standard and 8–10 are challenge questions. After you answer, the feedback shows the full solution.

Score: 0 / 10Answered: 0 / 10

Warm-up (The basic skill: one or two steps.)

1.6³ × 6⁴ =

Not quite. Multiplying powers: add the indices, don't multiply them.

Solution

  1. The base is the same (6) and we are multiplying, so add the indices.
  2. 3 + 4 = 7, so the answer is 6⁷.

Answer: C. 6⁷.

Not quite. Keep the base the same; don't multiply 6 × 6.

Solution

  1. The base is the same (6) and we are multiplying, so add the indices.
  2. 3 + 4 = 7, so the answer is 6⁷.

Answer: C. 6⁷.

Correct.

Solution

  1. The base is the same (6) and we are multiplying, so add the indices.
  2. 3 + 4 = 7, so the answer is 6⁷.

Not quite. Multiplying means add the indices, not subtract.

Solution

  1. The base is the same (6) and we are multiplying, so add the indices.
  2. 3 + 4 = 7, so the answer is 6⁷.

Answer: C. 6⁷.

2.5⁹ ÷ 5³ =

Not quite. Dividing powers: subtract the indices, don't divide them.

Solution

  1. The base is the same (5) and we are dividing, so subtract the indices.
  2. 9 − 3 = 6, so the answer is 5⁶.

Answer: B. 5⁶.

Correct.

Solution

  1. The base is the same (5) and we are dividing, so subtract the indices.
  2. 9 − 3 = 6, so the answer is 5⁶.

Not quite. Dividing means subtract, not add.

Solution

  1. The base is the same (5) and we are dividing, so subtract the indices.
  2. 9 − 3 = 6, so the answer is 5⁶.

Answer: B. 5⁶.

Not quite. Keep the base 5; don't divide 5 by 5.

Solution

  1. The base is the same (5) and we are dividing, so subtract the indices.
  2. 9 − 3 = 6, so the answer is 5⁶.

Answer: B. 5⁶.

3.(3²)⁴ =

Correct.

Solution

  1. A power of a power: multiply the indices.
  2. 2 × 4 = 8, so the answer is 3⁸.

Not quite. A power of a power: multiply the indices, don't add them.

Solution

  1. A power of a power: multiply the indices.
  2. 2 × 4 = 8, so the answer is 3⁸.

Answer: A. 3⁸.

Not quite. 3² is 9, not 6.

Solution

  1. A power of a power: multiply the indices.
  2. 2 × 4 = 8, so the answer is 3⁸.

Answer: A. 3⁸.

Not quite. The power 2 inside the bracket still counts.

Solution

  1. A power of a power: multiply the indices.
  2. 2 × 4 = 8, so the answer is 3⁸.

Answer: A. 3⁸.

Exam standard (Several steps, like real exam questions.)

4.7⁰ =

Not quite. 7⁰ is not 7 × 0. Any number to the power 0 is 1.

Solution

  1. 7³ ÷ 7³ = 1, and by the index law it is also 7⁰.
  2. So 7⁰ = 1. Any number (except 0) to the power 0 is 1.

Answer: C. 1.

Not quite. The index 0 does not leave the number unchanged.

Solution

  1. 7³ ÷ 7³ = 1, and by the index law it is also 7⁰.
  2. So 7⁰ = 1. Any number (except 0) to the power 0 is 1.

Answer: C. 1.

Correct.

Solution

  1. 7³ ÷ 7³ = 1, and by the index law it is also 7⁰.
  2. So 7⁰ = 1. Any number (except 0) to the power 0 is 1.

Not quite. The index is not a digit to write after 7.

Solution

  1. 7³ ÷ 7³ = 1, and by the index law it is also 7⁰.
  2. So 7⁰ = 1. Any number (except 0) to the power 0 is 1.

Answer: C. 1.

5.x⁴ × x⁶ =

Not quite. Add the indices, don't multiply them.

Solution

  1. Same base (x), multiplying: add the indices.
  2. 4 + 6 = 10, so the answer is x¹⁰.

Answer: C. x¹⁰.

Not quite. You are multiplying x by x, not adding two x terms.

Solution

  1. Same base (x), multiplying: add the indices.
  2. 4 + 6 = 10, so the answer is x¹⁰.

Answer: C. x¹⁰.

Correct.

Solution

  1. Same base (x), multiplying: add the indices.
  2. 4 + 6 = 10, so the answer is x¹⁰.

Not quite. Multiplying means add the indices.

Solution

  1. Same base (x), multiplying: add the indices.
  2. 4 + 6 = 10, so the answer is x¹⁰.

Answer: C. x¹⁰.

6.What is the value of 2⁶?

Correct.

Solution

  1. 2⁶ = 2 × 2 × 2 × 2 × 2 × 2.
  2. 2, 4, 8, 16, 32, 64: the answer is 64.

Not quite. 2⁶ means six 2s multiplied, not 2 × 6.

Solution

  1. 2⁶ = 2 × 2 × 2 × 2 × 2 × 2.
  2. 2, 4, 8, 16, 32, 64: the answer is 64.

Answer: A. 64.

Not quite. That is 6², not 2⁶.

Solution

  1. 2⁶ = 2 × 2 × 2 × 2 × 2 × 2.
  2. 2, 4, 8, 16, 32, 64: the answer is 64.

Answer: A. 64.

Not quite. That is 2⁵. Count six 2s.

Solution

  1. 2⁶ = 2 × 2 × 2 × 2 × 2 × 2.
  2. 2, 4, 8, 16, 32, 64: the answer is 64.

Answer: A. 64.

7.Work out 10³ × 10² as an ordinary number.

Not quite. 3 + 2 = 5, so 10⁵, not 10⁶.

Solution

  1. Add the indices: 10³ × 10² = 10⁵.
  2. 10⁵ is 1 followed by five zeros: 100,000.

Answer: D. 100,000.

Not quite. Multiply 1,000 by 100; don't add them.

Solution

  1. Add the indices: 10³ × 10² = 10⁵.
  2. 10⁵ is 1 followed by five zeros: 100,000.

Answer: D. 100,000.

Not quite. The index is not the answer: 10⁵ is 100,000.

Solution

  1. Add the indices: 10³ × 10² = 10⁵.
  2. 10⁵ is 1 followed by five zeros: 100,000.

Answer: D. 100,000.

Correct.

Solution

  1. Add the indices: 10³ × 10² = 10⁵.
  2. 10⁵ is 1 followed by five zeros: 100,000.

Challenge (Harder problems that mix ideas or need reasoning.)

8.Simplify 3⁴ × 3² ÷ 3³.

Not quite. Dividing by 3³ means subtract 3.

Solution

  1. Multiply first: add the indices, 4 + 2 = 6, giving 3⁶.
  2. Then divide: subtract, 6 − 3 = 3.
  3. The answer is 3³.

Answer: B. 3³.

Correct.

Solution

  1. Multiply first: add the indices, 4 + 2 = 6, giving 3⁶.
  2. Then divide: subtract, 6 − 3 = 3.
  3. The answer is 3³.

Not quite. Add 4 + 2 for the multiplying; don't multiply 4 × 2.

Solution

  1. Multiply first: add the indices, 4 + 2 = 6, giving 3⁶.
  2. Then divide: subtract, 6 − 3 = 3.
  3. The answer is 3³.

Answer: B. 3³.

Not quite. Keep the base 3; only the indices change.

Solution

  1. Multiply first: add the indices, 4 + 2 = 6, giving 3⁶.
  2. Then divide: subtract, 6 − 3 = 3.
  3. The answer is 3³.

Answer: B. 3³.

9.Which of these cannot be written as a single power?

Not quite. Same base: 5⁶.

Solution

  1. The index laws only work when the bases are the same.
  2. In 2³ × 3² the bases are 2 and 3, so it cannot be written as one power.

Answer: D. 2³ × 3².

Not quite. Same base (7 = 7¹): 7⁵.

Solution

  1. The index laws only work when the bases are the same.
  2. In 2³ × 3² the bases are 2 and 3, so it cannot be written as one power.

Answer: D. 2³ × 3².

Not quite. Power of a power: 4⁶.

Solution

  1. The index laws only work when the bases are the same.
  2. In 2³ × 3² the bases are 2 and 3, so it cannot be written as one power.

Answer: D. 2³ × 3².

Correct.

Solution

  1. The index laws only work when the bases are the same.
  2. In 2³ × 3² the bases are 2 and 3, so it cannot be written as one power.

10.Simplify 3x² × 4x⁵.

Not quite. Multiply the numbers: 3 × 4 = 12, not 3 + 4.

Solution

  1. Multiply the numbers: 3 × 4 = 12.
  2. Multiply the powers of x: add the indices, 2 + 5 = 7.
  3. So the answer is 12x⁷.

Answer: B. 12x⁷.

Correct.

Solution

  1. Multiply the numbers: 3 × 4 = 12.
  2. Multiply the powers of x: add the indices, 2 + 5 = 7.
  3. So the answer is 12x⁷.

Not quite. Add the indices of x: 2 + 5 = 7.

Solution

  1. Multiply the numbers: 3 × 4 = 12.
  2. Multiply the powers of x: add the indices, 2 + 5 = 7.
  3. So the answer is 12x⁷.

Answer: B. 12x⁷.

Not quite. Multiply the numbers and add the indices.

Solution

  1. Multiply the numbers: 3 × 4 = 12.
  2. Multiply the powers of x: add the indices, 2 + 5 = 7.
  3. So the answer is 12x⁷.

Answer: B. 12x⁷.

Apply

Computer memory

Computers count in twos. 1 kilobyte (KB) = 2¹⁰ bytes, 1 megabyte (MB) = 2¹⁰ KB, and 1 gigabyte (GB) = 2¹⁰ MB.

1. How many bytes are in 1 MB? Give your answer as a power of 2.

2. How many bytes are in 1 GB, as a power of 2?

3. A short message uses 2⁵ bytes. How many messages fit in 1 KB? Give your answer as a power of 2 and as a number.

  1. 2¹⁰ × 2¹⁰ = 2²⁰ bytes.
  2. 2²⁰ × 2¹⁰ = 2³⁰ bytes.
  3. 2¹⁰ ÷ 2⁵ = 2⁵ = 32 messages.

Next: the Lesson 2 exit quiz. Five questions, recorded on your dashboard. Aim for 4 out of 5.

PreviousNext