Powers · What is a power? · 1.4

Coefficients & powers.

A term like 3x² has a coefficient (3) and a power (x²). It means 3·x·x — the coefficient multiplies, and only x is raised. The exponent never touches the coefficient.

01 / THE IDEA

See it work.

Watch the method

Coefficients & powers

3·x·x is the coefficient 3 times x² = 3x².

1 / 2
3⋅x⋅x3 \cdot x \cdot x
1

Count the equal factors

Each repeated base becomes an exponent — the count of how many times it appears.

3⋅x⋅x3 \cdot x \cdot x
2

Write as a power

The coefficient stays in front; the repeated variable becomes a power.

=3x2= 3 x^{2}
— / WHY IT WORKS

The idea underneath.

Coefficient × power

The number in front multiplies the power. 3x² = 3·x·x. Only the variable carries the exponent here.

Why it works

5x³ = 5·x·x·x.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Write as a single power: z · z

1

Count the equal factors

Each repeated base becomes an exponent — the count of how many times it appears. z · z

2

Write as a power

The coefficient stays in front; the repeated variable becomes a power. = z^2

EX 2medium

Write as a single power: 3 · z · z · z · z

1

Count the equal factors

Each repeated base becomes an exponent — the count of how many times it appears. 3 · z · z · z · z

2

Write as a power

The coefficient stays in front; the repeated variable becomes a power. = 3 z^4

EX 3hard

Write as a single power: 3 · z · z

1

Count the equal factors

Each repeated base becomes an exponent — the count of how many times it appears. 3 · z · z

2

Write as a power

The coefficient stays in front; the repeated variable becomes a power. = 3 z^2

04 / COMMON TRAPS

Where students slip.

Trap

Thinking 3x² = (3x)².

Fix

3x² = 3·x·x = 3x²; (3x)² = 9x². The exponent only hits x here.

Trap

Multiplying the coefficient into the exponent.

Fix

The coefficient stays in front.

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

Write as a single power: 2 · x · x · x

Need a hint?
  • The number in front just multiplies.
  • Only the variable is raised.
C2

Write as a single power: 2 · z · z · z · z

Need a hint?
  • The number in front just multiplies.
  • Only the variable is raised.
C3

Write as a single power: 3 · x · x · x · x

Need a hint?
  • The number in front just multiplies.
  • Only the variable is raised.
C4

Write as a single power: 2 · z · z

Need a hint?
  • The number in front just multiplies.
  • Only the variable is raised.
Apply

Apply (4)

A1

Write as a single power: 3 · y · y

Need a hint?
  • The number in front just multiplies.
  • Only the variable is raised.
A2

Write as a single power: 2 · y · y · y

Need a hint?
  • The number in front just multiplies.
  • Only the variable is raised.
A3

Write as a single power: 3 · z · z

Need a hint?
  • The number in front just multiplies.
  • Only the variable is raised.
A4

Write as a single power: y · y

Need a hint?
  • The number in front just multiplies.
  • Only the variable is raised.
Extension

Extension (4)

E1

Write as a single power: 2 · y · y · y

Need a hint?
  • The number in front just multiplies.
  • Only the variable is raised.
E2

Write as a single power: 2 · x · x

Need a hint?
  • The number in front just multiplies.
  • Only the variable is raised.
E3

Write as a single power: y · y · y

Need a hint?
  • The number in front just multiplies.
  • Only the variable is raised.
E4

Write as a single power: z · z · z · z

Need a hint?
  • The number in front just multiplies.
  • Only the variable is raised.