Powers · Power of a power · 5.1

Power of a power.

Raising a power to a power MULTIPLIES the exponents: (xᵃ)ᵇ = xᵃᵇ. (x³)⁴ = x³ four times = x¹². This is the rule people confuse with the product rule.

01 / THE IDEA

See it work.

Watch the method

Power of a power

(x³)⁴ = x¹² (3 × 4).

1 / 1
(x3)4(x^{3})^{4}
1

Multiply the exponents

A power of a power: keep the base, multiply 3 × 4.

(x3)4=x12(x^{3})^{4} = x^{12}
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type with ^ for powers — e.g. x^5.

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… =
— / WHY IT WORKS

The idea underneath.

Multiply, not add

(x³)⁴ means x³ multiplied 4 times: x³·x³·x³·x³ = x¹². So you multiply the exponents.

Why it works

Four groups of three x’s = twelve x’s.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify (x^4)^2

1

Multiply the exponents

A power of a power: keep the base, multiply 4 × 2. (x^4)^2 = x^8

EX 2medium

Simplify (x^4)^3

1

Multiply the exponents

A power of a power: keep the base, multiply 4 × 3. (x^4)^3 = x^12

EX 3hard

Simplify (x^2)^3

1

Multiply the exponents

A power of a power: keep the base, multiply 2 × 3. (x^2)^3 = x^6

04 / COMMON TRAPS

Where students slip.

Trap

Adding the exponents (getting x⁷).

Fix

Power of a power MULTIPLIES; multiplying powers ADDS.

Trap

Forgetting to raise the whole base.

Fix

The outer power hits the entire inside.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Simplify (z^2)^4

Need a hint?
  • Multiply the two exponents.
  • Don’t confuse it with the product rule.
C2

Simplify (x^3)^2

Need a hint?
  • Multiply the two exponents.
  • Don’t confuse it with the product rule.
C3

Simplify (x^2)^4

Need a hint?
  • Multiply the two exponents.
  • Don’t confuse it with the product rule.
C4

Simplify (x^2)^4

Need a hint?
  • Multiply the two exponents.
  • Don’t confuse it with the product rule.
Apply

Apply (4)

A1

Simplify (x^3)^3

Need a hint?
  • Multiply the two exponents.
  • Don’t confuse it with the product rule.
A2

Simplify (z^4)^2

Need a hint?
  • Multiply the two exponents.
  • Don’t confuse it with the product rule.
A3

Simplify (x^2)^3

Need a hint?
  • Multiply the two exponents.
  • Don’t confuse it with the product rule.
A4

Simplify (y^2)^3

Need a hint?
  • Multiply the two exponents.
  • Don’t confuse it with the product rule.
Extension

Extension (4)

E1

Simplify (z^3)^3

Need a hint?
  • Multiply the two exponents.
  • Don’t confuse it with the product rule.
E2

Simplify (x^3)^3

Need a hint?
  • Multiply the two exponents.
  • Don’t confuse it with the product rule.
E3

Simplify (x^3)^2

Need a hint?
  • Multiply the two exponents.
  • Don’t confuse it with the product rule.
E4

Simplify (z^4)^3

Need a hint?
  • Multiply the two exponents.
  • Don’t confuse it with the product rule.