Powers · Power of a power · 5.2

Power of a product.

A power of a product raises EVERY factor — including the coefficient. (3x)² = 3²x² = 9x². This differs from 3x², where only x is squared.

01 / THE IDEA

See it work.

Watch the method

Power of a product

(3x)² = 9x² — the 3 is squared too.

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

Raise EACH factor

A power of a product hits every factor — including the coefficient.

(3x)2(3x)^{2}
2

Apply the power

Raise each factor to the outside power.

=9x2= 9 x^{2}
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.

Everything gets the power

(3x)² = (3x)(3x) = 9x². Compare 3x² = 3·x·x. The parentheses make all the difference.

Why it works

(3x)² squares the 3 AND the x.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify (4z)^3

1

Raise EACH factor

A power of a product hits every factor — including the coefficient. (4z)^3

2

Apply the power

Raise each factor to the outside power. = 64 z^3

EX 2medium

Simplify (3x)^2

1

Raise EACH factor

A power of a product hits every factor — including the coefficient. (3x)^2

2

Apply the power

Raise each factor to the outside power. = 9 x^2

EX 3hard

Simplify (xy)^3

1

Raise EACH factor

A power of a product hits every factor — including the coefficient. (xy)^3

2

Apply the power

Raise each factor to the outside power. = x^3 y^3

04 / COMMON TRAPS

Where students slip.

Trap

Writing (3x)² = 3x².

Fix

It is 9x² — the coefficient is squared too.

Trap

Forgetting the coefficient.

Fix

Raise every factor inside the brackets.

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 (xy)^3

Need a hint?
  • Raise each factor — number and variable.
  • Mind the difference from 3x².
C2

Simplify (xy)^2

Need a hint?
  • Raise each factor — number and variable.
  • Mind the difference from 3x².
C3

Simplify (xy)^2

Need a hint?
  • Raise each factor — number and variable.
  • Mind the difference from 3x².
C4

Simplify (3x)^3

Need a hint?
  • Raise each factor — number and variable.
  • Mind the difference from 3x².
Apply

Apply (4)

A1

Simplify (3x)^3

Need a hint?
  • Raise each factor — number and variable.
  • Mind the difference from 3x².
A2

Simplify (4z)^2

Need a hint?
  • Raise each factor — number and variable.
  • Mind the difference from 3x².
A3

Simplify (xy)^2

Need a hint?
  • Raise each factor — number and variable.
  • Mind the difference from 3x².
A4

Simplify (xy)^3

Need a hint?
  • Raise each factor — number and variable.
  • Mind the difference from 3x².
Extension

Extension (4)

E1

Simplify (3y)^2

Need a hint?
  • Raise each factor — number and variable.
  • Mind the difference from 3x².
E2

Simplify (xy)^2

Need a hint?
  • Raise each factor — number and variable.
  • Mind the difference from 3x².
E3

Simplify (2z)^3

Need a hint?
  • Raise each factor — number and variable.
  • Mind the difference from 3x².
E4

Simplify (xy)^2

Need a hint?
  • Raise each factor — number and variable.
  • Mind the difference from 3x².