Powers · Zero & negative exponents · 6.2

Negative exponents.

A negative exponent means the reciprocal: x⁻ⁿ = 1/xⁿ. x⁻³ = 1/x³. The negative sign moves the factor to the denominator and makes the exponent positive.

01 / THE IDEA

See it work.

Watch the method

Negative exponents

x⁻³ = 1/x³.

1 / 1
x−3x^{-3}
1

Negative exponent

A negative exponent means the reciprocal: move it to the denominator and make the exponent positive.

x−3=1x3x^{-3} = \frac{1}{x^{3}}
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.

Reciprocal

Counting down the exponents past 0 keeps dividing by x: x², x¹, x⁰=1, x⁻¹ = 1/x, x⁻² = 1/x². A negative exponent is a denominator.

Why it works

x⁻² = 1/x².

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Rewrite with positive exponents: y^-4

1

Negative exponent

A negative exponent means the reciprocal: move it to the denominator and make the exponent positive. y^-4 = 1/y^4

EX 2medium

Rewrite with positive exponents: x^-3

1

Negative exponent

A negative exponent means the reciprocal: move it to the denominator and make the exponent positive. x^-3 = 1/x^3

EX 3hard

Rewrite with positive exponents: z^-4

1

Negative exponent

A negative exponent means the reciprocal: move it to the denominator and make the exponent positive. z^-4 = 1/z^4

04 / COMMON TRAPS

Where students slip.

Trap

Treating x⁻³ as a negative number.

Fix

It means 1/x³, not −x³.

Trap

Leaving the exponent negative.

Fix

Move it to the denominator and make it positive.

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

Rewrite with positive exponents: x^-2

Need a hint?
  • Move it to the denominator.
  • Make the exponent positive.
C2

Rewrite with positive exponents: y^-4

Need a hint?
  • Move it to the denominator.
  • Make the exponent positive.
C3

Rewrite with positive exponents: x^-5

Need a hint?
  • Move it to the denominator.
  • Make the exponent positive.
C4

Rewrite with positive exponents: x^-4

Need a hint?
  • Move it to the denominator.
  • Make the exponent positive.
Apply

Apply (4)

A1

Rewrite with positive exponents: y^-2

Need a hint?
  • Move it to the denominator.
  • Make the exponent positive.
A2

Rewrite with positive exponents: y^-4

Need a hint?
  • Move it to the denominator.
  • Make the exponent positive.
A3

Rewrite with positive exponents: x^-2

Need a hint?
  • Move it to the denominator.
  • Make the exponent positive.
A4

Rewrite with positive exponents: y^-4

Need a hint?
  • Move it to the denominator.
  • Make the exponent positive.
Extension

Extension (4)

E1

Rewrite with positive exponents: y^-2

Need a hint?
  • Move it to the denominator.
  • Make the exponent positive.
E2

Rewrite with positive exponents: x^-5

Need a hint?
  • Move it to the denominator.
  • Make the exponent positive.
E3

Rewrite with positive exponents: x^-4

Need a hint?
  • Move it to the denominator.
  • Make the exponent positive.
E4

Rewrite with positive exponents: x^-5

Need a hint?
  • Move it to the denominator.
  • Make the exponent positive.