Power of a Power

Learn why a power raised to a power means multiplying the exponents, and how to raise products, quotients and coefficients to a power.

Intermediate25 min18 problems
Practice Problems

Definition

When a power is raised to another power, you keep the base and multiply the exponents:
(am)n=am⋅n(a^m)^n = a^{m \cdot n}
Why it works: the outer exponent nn says "use ama^m as a factor nn times". Then the product rule (add the exponents) does the rest:
(23)2=23⋅23=23+3=26=64(2^3)^2 = 2^3 \cdot 2^3 = 2^{3+3} = 2^6 = 64
Adding the exponent mm a total of nn times is the same as m⋅nm \cdot n.
Two related rules come from the same idea:
  • Power of a product: (ab)n=anbn(ab)^n = a^n b^n. Every factor inside the brackets gets the exponent, also the number in front: (3x2)3=33⋅(x2)3=27x6(3x^2)^3 = 3^3 \cdot (x^2)^3 = 27x^6
  • Power of a quotient: (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} (with b≠0b \neq 0)
Watch the brackets: (23)2=26=64(2^3)^2 = 2^6 = 64, but 232=29=5122^{3^2} = 2^9 = 512. Without brackets, the exponent at the top is worked out first.
In this lesson all exponents are positive whole numbers.

Try it now

Simplify (x3)4(x^3)^4.

Worked Examples

Simplify (52)4(5^2)^4.
1

Find the base and the two exponents

Base 55, inner exponent 22, outer exponent 44 → A power raised to a power
2

Write it as a product

(52)4=52⋅52⋅52⋅52=52+2+2+2(5^2)^4 = 5^2 \cdot 5^2 \cdot 5^2 \cdot 5^2 = 5^{2+2+2+2} → 585^8
3

Use the shortcut: multiply the exponents

2⋅4=82 \cdot 4 = 8 → (52)4=58(5^2)^4 = 5^8

Common Mistakes

Adding the exponents: (x3)4=x7(x^3)^4 = x^7

Why it's wrong: Adding is the product rule, for x3⋅x4x^3 \cdot x^4. Here the outer 44 means four copies of x3x^3, so the exponent is 3+3+3+3=123 + 3 + 3 + 3 = 12.

Correct: Multiply the exponents: (x3)4=x3⋅4=x12(x^3)^4 = x^{3 \cdot 4} = x^{12}

Thinking (a+b)2=a2+b2(a + b)^2 = a^2 + b^2

Why it's wrong: The rule (ab)n=anbn(ab)^n = a^n b^n is for a product, not a sum. Test with numbers: (3+4)2=72=49(3 + 4)^2 = 7^2 = 49, but 32+42=9+16=253^2 + 4^2 = 9 + 16 = 25.

Correct: (a+b)2=(a+b)(a+b)=a2+2ab+b2(a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2, so (3+4)2=9+24+16=49(3 + 4)^2 = 9 + 24 + 16 = 49

Treating (23)2(2^3)^2 and 2322^{3^2} as the same

Why it's wrong: The brackets change the order. (23)2(2^3)^2 squares 232^3, while 2322^{3^2} first works out the top exponent 32=93^2 = 9.

Correct: (23)2=26=64(2^3)^2 = 2^6 = 64, but 232=29=5122^{3^2} = 2^9 = 512

Forgetting the coefficient: (3x2)3=3x6(3x^2)^3 = 3x^6

Why it's wrong: The exponent applies to every factor inside the brackets, and the number 33 is a factor too.

Correct: (3x2)3=33⋅x6=27x6(3x^2)^3 = 3^3 \cdot x^6 = 27x^6

Interactive Visual

Exponents as repeated factors

Power of a power

Power of a power: multiply the exponents.

(23)2\left(2^{3}\right)^{2}

Write each power as repeated factors

22·22·22
232^{3}3 factors
22·22·22
232^{3}3 factors

2 groups of 3: 2 · 3 = 6 factors

Result

(23)2\displaystyle \left(2^{3}\right)^{2}==23⋅2\displaystyle 2^{3 \cdot 2}==26\displaystyle 2^{6}

Value: 26=642^{6} = 64

Order of Operations

  1. 1. Parentheses
  2. 2. Exponents
  3. 3. × and ÷
  4. 4. + and −
2322^{\colorbox{#fde68a}{$\color{#1f2937}3^{2}$}}

Exponents next: 32=93^{2} = 9

Step 1 of 2

Area model: square of a binomial

A square with side x + a: two equal a·x strips and an a² corner, so the middle term is 2ax.

x2x^2
3x3x
3x3x
99
xx
33
xx
33

Length × width = area

(x+3)2=x2+6x+9(x + 3)^2 = x^2 + 6x + 9

Interactive Sandbox

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Practice Problems

18 problems
Problem 1 of 18
Easy
Simplify (x3)4(x^3)^4.

Why It Matters

The power of a power rule turns long chains of multiplication into one quick step:
  • Areas and volumes: a square with side 10310^3 m has area (103)2=106(10^3)^2 = 10^6 m², and a cube with edge 222^2 cm has volume (22)3=26=64(2^2)^3 = 2^6 = 64 cm³
  • Computer storage: a kilobyte is 2102^{10} bytes, and (210)3=230(2^{10})^3 = 2^{30} bytes is a little more than one billion bytes
  • Science: in E=mc2E = mc^2 the speed of light 3⋅1083 \cdot 10^8 m/s is squared: (3⋅108)2=9⋅1016(3 \cdot 10^8)^2 = 9 \cdot 10^{16}
  • Algebra: a square with side 3x23x^2 has area (3x2)2=9x4(3x^2)^2 = 9x^4
Together with the product rule and the quotient rule, it lets you simplify almost any expression with powers of the same base.

Real World Applications

Computer Storage

Computers count memory in powers of 2. Each bigger unit is $2^{10} = 1024$ times the one before it.

Example:

A kilobyte is 2102^{10} bytes, a megabyte is 2102^{10} kilobytes, so a megabyte is (210)2=220(2^{10})^2 = 2^{20} bytes.

1Try It Yourself

A gigabyte is 2102^{10} megabytes, so it is 210⋅210⋅2102^{10} \cdot 2^{10} \cdot 2^{10} bytes, which is (210)3(2^{10})^3 bytes.

Write the number of bytes in a gigabyte as a single power of 2.

Step 1: Write the mathematical expression

Simplify (210)3(2^{10})^3:

Enlarging a Photo

When a picture is enlarged, its width and height are both multiplied by the same factor, so its area (the number of pixels) grows by that factor squared.

Example:

Doubling the width and the height of a photo multiplies the number of pixels by 22=42^2 = 4.

2Try It Yourself

A photo is enlarged so that its width and its height are each multiplied by 232^3.

By what factor does the number of pixels grow?

Step 1: Write the mathematical expression

The area factor is (23)2(2^3)^2:

Energy and the Speed of Light

Einstein's formula $E = mc^2$ squares the speed of light $c$, which is about $3 \cdot 10^8$ meters per second.

Example:

To square 3⋅1083 \cdot 10^8, square the 33 and the power of ten separately, because both are factors.

3Try It Yourself

The speed of light is about c=3⋅108c = 3 \cdot 10^8 m/s.

What is c2=(3⋅108)2c^2 = (3 \cdot 10^8)^2 in scientific notation?

Step 1: Write the mathematical expression

Use the power of a product and the power of a power:

Key Takeaways

  • 1Power of a power: (am)n=am⋅n(a^m)^n = a^{m \cdot n}, keep the base and multiply the exponents
  • 2The rule comes from the product rule: (23)2=23⋅23=23+3=26(2^3)^2 = 2^3 \cdot 2^3 = 2^{3+3} = 2^6
  • 3Power of a product: (ab)n=anbn(ab)^n = a^n b^n, every factor gets the exponent, including the coefficient: (3x2)3=27x6(3x^2)^3 = 27x^6
  • 4Power of a quotient: (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, raise the numerator and the denominator
  • 5Brackets matter: (23)2=64(2^3)^2 = 64, but 232=5122^{3^2} = 512
  • 6The rules work for products and quotients only: (a+b)2≠a2+b2(a + b)^2 \neq a^2 + b^2

Frequently Asked Questions

Because (am)n(a^m)^n means nn copies of ama^m multiplied together. By the product rule their exponents add up: m+m+⋯+mm + m + \dots + m (nn times), which is m⋅nm \cdot n.
Because (am)n(a^m)^n means nn copies of ama^m multiplied together. By the product rule their exponents add up: m+m+⋯+mm + m + \dots + m (nn times), which is m⋅nm \cdot n.
Yes. Both equal a6a^6, because 2⋅3=3⋅22 \cdot 3 = 3 \cdot 2. Check with numbers: (22)3=43=64(2^2)^3 = 4^3 = 64 and (23)2=82=64(2^3)^2 = 8^2 = 64.
A tower of exponents is worked out from the top down: first 32=93^2 = 9, then 29=5122^9 = 512. If you mean (23)2=64(2^3)^2 = 64, you must write the brackets.
No. The exponent can be shared out only over factors. For a sum, multiply it out: (x+2)2=(x+2)(x+2)=x2+4x+4(x + 2)^2 = (x + 2)(x + 2) = x^2 + 4x + 4.

Glossary

Power of a power
A power raised to another power, like (am)n(a^m)^n. It equals am⋅na^{m \cdot n}.
Power of a product
A product raised to a power: (ab)n=anbn(ab)^n = a^n b^n. Every factor gets the exponent.
Power of a quotient
A fraction raised to a power: (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, with b≠0b \neq 0.
Coefficient
The number in front of a variable. In 3x23x^2 the coefficient is 33.
Base
The number or variable that is multiplied. In (x2)3(x^2)^3 the base is xx.
Exponent
The small raised number that says how many times the base is used as a factor.

Formula Card

Power of a power

(am)n=am⋅n(a^m)^n = a^{m \cdot n}

Keep the base, multiply the exponents

Power of a product

(ab)n=anbn(ab)^n = a^n b^n

Every factor in the brackets gets the exponent

Power of a quotient

(ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

Raise the numerator and the denominator ($b \neq 0$)

Today's challengeOne problem, three tries

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