Year 9 · Unit 1 Real numbers and indices · Lesson 4 of 7

A power of a power: multiply the exponents

Raise a power to a power and the exponents multiply. Raise a product to a power and every factor takes the power. Raise a sum to a power and nothing splits at all.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. (Lesson 3) Write 54 × 56 as one power.
    Answer510, because the exponents add: 4 + 6 = 10.
  2. (Year 8, Unit 3, lesson 5) What is (−3)2?
    Answer9, because two negative factors multiply to a positive.
  3. (Year 7, Unit 1, lesson 6) Work out 3 + 4 × 22.
    Answer19: power first, 22 = 4, then 4 × 4 = 16, then 3 + 16 = 19.
Words used in this lesson
power of a power
a power that is itself raised to a power. For example: (23)2.
bracket
a pair of curved lines that group terms. For example: (−2)4.
negative base
a base that is below zero. For example: −2 in (−2)4.
even power
a power whose exponent is even. For example: 4 in (−2)4.
odd power
a power whose exponent is odd. For example: 5 in (−2)5.
expand
write out a power or a product in full. For example: 23 = 2 × 2 × 2.

A power of a power

What does (23)2 mean? It means two copies of 23 multiplied together: 23 × 23. Predict first: is (23)2 equal to 25 or to 26? Then set the base and the two exponents and see how many factors there are in all.

3
2
3
3⁶as one power
729value
2 × 3 = 6exponents

(3²)³ = 3⁶ = 729: 3 rows of 2 factors make 2 × 3 = 6 factors of 3.

Show me the steps
  1. (2³)² = 2⁶ = 64: 2 rows of 3 factors make 3 × 2 = 6 factors of 2.
  2. (3²)³ = 3⁶ = 729: 3 rows of 2 factors make 2 × 3 = 6.
  3. (5¹)⁴ = 5⁴ = 625: 4 rows of 1 factor make 1 × 4 = 4.
  4. (2⁴)⁴ = 2¹⁶ = 65536: 4 rows of 4 factors make 4 × 4 = 16.
  5. (2⁵)³ = 2¹⁵ = 32768: 3 rows of 5 factors make 5 × 3 = 15.
  6. (4⁵)² = 4¹⁰ = 1048576: 2 rows of 5 factors make 5 × 2 = 10.

A power of a power: (am)n = am × n. There are n rows of m factors, so the number of factors is m times n. Compare the product law from Lesson 3, which adds exponents because it joins two runs. A power of a power repeats a run, so it multiplies. So (23)2 = 23 × 2 = 26 = 64, not 25.

A power of a product

(2 × 3)2 means (2 × 3) × (2 × 3). The factors can be regrouped in any order, so that is 2 × 2 × 3 × 3 = 22 × 32. Every factor inside the bracket takes the power. A quotient works the same way: (2/3)3 = 23/33 = 8/27.

2
5
2
100(a × b)ⁿ and aⁿ × bⁿ
49(a + b)ⁿ
29aⁿ + bⁿ

(2 × 5)² = 10² = 100 and 2² × 5² = 4 × 25 = 100: equal. (2 + 5)² = 49 but 2² + 5² = 4 + 25 = 29: not equal.

Show me the steps
  1. (2 × 3)² = 6² = 36 = 2² × 3² = 4 × 9. But (2 + 3)² = 25, not 2² + 3² = 13.
  2. (2 × 5)³ = 10³ = 1000 = 2³ × 5³ = 8 × 125. But (2 + 5)³ = 343, not 8 + 125 = 133.
  3. (3 × 4)² = 12² = 144 = 3² × 4² = 9 × 16. But (3 + 4)² = 49, not 9 + 16 = 25.
  4. (2 × 6)⁴ = 12⁴ = 20736 = 2⁴ × 6⁴ = 16 × 1296. But (2 + 6)⁴ = 4096, not 16 + 1296 = 1312.
  5. (5 × 5)¹ = 25 = 5¹ × 5¹. With exponent 1 even the sum agrees, 10 = 5 + 5, because nothing is raised.

A power of a product: (ab)n = anbn. And a power of a quotient: (a/b)n = an/bn.

A common mistake: splitting a sum. (3 + 4)2 is 7 × 7 = 49, not 32 + 42 = 9 + 16 = 25. The laws work for multiplication and division inside the bracket. They do not work for addition or subtraction: (a + b)2 also has the middle term 2ab.

Negative bases

A negative base needs its brackets. (−3)4 = (−3) × (−3) × (−3) × (−3) = 81, because the negative factors cancel in pairs. Without brackets, −34 means −(34) = −81: the power touches only the 3.

Worked example: a number and a letter

Simplify (2x3)4.

  1. Every factor in the bracket takes the power 4: 24 × (x3)4.
  2. 24 = 16, and (x3)4 = x3 × 4 = x12.
  3. 16x12. The 2 is raised to the power too, so it becomes 16, not 2.

Write (34)2 as one power and find its value.

  1. Multiply the exponents: 4 × 2 = 8.
  2. (34)2 = 38 = 6561. Check: 34 = 81 and 81 × 81 = 6561.

Your go. The first step is given. Simplify (5y2)3.

  1. Every factor takes the power 3: 53 × (y2)3.
  2. Work out each part, then put them together and check.
Check the answer

53 = 125 and (y2)3 = y6, so the answer is 125y6.

Check before you go on

1. Which is (43)2 written as a power of 4?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A square tile has side 24 cm. Its area is 2k cm2. What is k?

3. Evaluate (−2)4.

Level 2 · Building

4. (3a2/b)3 = 27ak/b3. What is k?

5. Work out (4 + 5)2 − (42 + 52).

6. Which of these equals −16?

Level 3 · Stretch

7. (2x3y2)3 = 8xayb. What is a + b?

8. (2k)3 = 46. What is k?

Something to think about

Work out (23)2. Then work out 2 raised to the power 32, which is 2 to the power 9. Are the two answers equal?

One way to think about it

(23)2 = 82 = 64, but 2 to the power 9 is 29 = 512. The brackets change what is raised to what, so they change the answer: 23 squared multiplies the exponents (3 × 2 = 6), and 2 to the power 32 raises the exponent itself (32 = 9).

Exit ticket

9. Which of these is NOT equal to 212?