Powers · Power of a power · 5.4

Combine the rules.

Put the rules together: (2x²)³ raises the coefficient (2³ = 8) AND multiplies the inner exponent (2·3 = 6), giving 8x⁶.

01 / THE IDEA

See it work.

Watch the method

Combine the rules

(2x²)³ = 8x⁶.

1 / 1
(2x2)3(2x^{2})^{3}
1

Multiply the exponents

Power of a power: multiply the exponents.

(2x2)3=8x6(2x^{2})^{3} = 8 x^{6}
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type with ^ for powers — e.g. x^5.

streak 0
best 0
… =
— / WHY IT WORKS

The idea underneath.

Both at once

Power of a product (the 2 gets cubed) plus power of a power (the x² becomes x⁶).

Why it works

2³ = 8 and (x²)³ = x⁶.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify (3z^3)^2

1

Multiply the exponents

Power of a power: multiply the exponents. (3z^3)^2 = 9 z^6

EX 2medium

Simplify (3y^2)^3

1

Multiply the exponents

Power of a power: multiply the exponents. (3y^2)^3 = 27 y^6

EX 3hard

Simplify (3y^2)^3

1

Multiply the exponents

Power of a power: multiply the exponents. (3y^2)^3 = 27 y^6

04 / COMMON TRAPS

Where students slip.

Trap

Forgetting to cube the 2.

Fix

(2x²)³ = 2³x⁶ = 8x⁶.

Trap

Adding the exponents.

Fix

Power of a power multiplies: 2·3 = 6.

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 (2y^3)^2

Need a hint?
  • Raise the coefficient to the outer power.
  • Multiply the inner exponent.
C2

Simplify (3z^2)^3

Need a hint?
  • Raise the coefficient to the outer power.
  • Multiply the inner exponent.
C3

Simplify (2z^3)^2

Need a hint?
  • Raise the coefficient to the outer power.
  • Multiply the inner exponent.
C4

Simplify (3x^3)^2

Need a hint?
  • Raise the coefficient to the outer power.
  • Multiply the inner exponent.
Apply

Apply (4)

A1

Simplify (3x^2)^3

Need a hint?
  • Raise the coefficient to the outer power.
  • Multiply the inner exponent.
A2

Simplify (3x^3)^2

Need a hint?
  • Raise the coefficient to the outer power.
  • Multiply the inner exponent.
A3

Simplify (3x^3)^3

Need a hint?
  • Raise the coefficient to the outer power.
  • Multiply the inner exponent.
A4

Simplify (3x^3)^3

Need a hint?
  • Raise the coefficient to the outer power.
  • Multiply the inner exponent.
Extension

Extension (4)

E1

Simplify (3y^3)^2

Need a hint?
  • Raise the coefficient to the outer power.
  • Multiply the inner exponent.
E2

Simplify (2x^2)^3

Need a hint?
  • Raise the coefficient to the outer power.
  • Multiply the inner exponent.
E3

Simplify (2y^2)^2

Need a hint?
  • Raise the coefficient to the outer power.
  • Multiply the inner exponent.
E4

Simplify (2x^3)^2

Need a hint?
  • Raise the coefficient to the outer power.
  • Multiply the inner exponent.