Powers · Zero & negative exponents · 6.4

Laws making negatives.

When the quotient rule gives a negative exponent, finish by writing it with a positive exponent in the denominator. x² ÷ x⁵ = x⁻³ = 1/x³.

01 / THE IDEA

See it work.

Watch the method

Laws making negatives

x² ÷ x⁵ = x⁻³ = 1/x³.

1 / 2
x2x5\frac{x^{2}}{x^{5}}
1

Spot the negative exponent

A negative exponent belongs in the denominator.

x2x5\frac{x^{2}}{x^{5}}
2

Rewrite with positive exponents

Flip the negative-exponent factors to the bottom.

=1x3= \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 or 1/x^2.

streak 0
best 0
… =
— / WHY IT WORKS

The idea underneath.

Finish the job

Apply the quotient rule, then convert any negative exponent to a positive one in the denominator.

Why it works

x²/x⁵ = 1/x³.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify z^2 ÷ z^4

1

Spot the negative exponent

A negative exponent belongs in the denominator. \fracz^2z^4

2

Rewrite with positive exponents

Flip the negative-exponent factors to the bottom. = 1/z^2

EX 2medium

Simplify z^1 ÷ z^2

1

Spot the negative exponent

A negative exponent belongs in the denominator. \fracz^1z^2

2

Rewrite with positive exponents

Flip the negative-exponent factors to the bottom. = 1/z

EX 3hard

Simplify y^1 ÷ y^2

1

Spot the negative exponent

A negative exponent belongs in the denominator. \fracy^1y^2

2

Rewrite with positive exponents

Flip the negative-exponent factors to the bottom. = 1/y

04 / COMMON TRAPS

Where students slip.

Trap

Stopping at x⁻³.

Fix

Finish: x⁻³ = 1/x³.

Trap

Writing −x³.

Fix

A negative exponent is a denominator, not a negative number.

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^1 ÷ z^2

Need a hint?
  • Subtract the exponents first.
  • Then rewrite as 1/xⁿ.
C2

Simplify x^1 ÷ x^4

Need a hint?
  • Subtract the exponents first.
  • Then rewrite as 1/xⁿ.
C3

Simplify x^1 ÷ x^4

Need a hint?
  • Subtract the exponents first.
  • Then rewrite as 1/xⁿ.
C4

Simplify z^1 ÷ z^4

Need a hint?
  • Subtract the exponents first.
  • Then rewrite as 1/xⁿ.
Apply

Apply (4)

A1

Simplify x^2 ÷ x^3

Need a hint?
  • Subtract the exponents first.
  • Then rewrite as 1/xⁿ.
A2

Simplify z^1 ÷ z^4

Need a hint?
  • Subtract the exponents first.
  • Then rewrite as 1/xⁿ.
A3

Simplify y^1 ÷ y^3

Need a hint?
  • Subtract the exponents first.
  • Then rewrite as 1/xⁿ.
A4

Simplify y^2 ÷ y^3

Need a hint?
  • Subtract the exponents first.
  • Then rewrite as 1/xⁿ.
Extension

Extension (4)

E1

Simplify y^2 ÷ y^5

Need a hint?
  • Subtract the exponents first.
  • Then rewrite as 1/xⁿ.
E2

Simplify z^1 ÷ z^2

Need a hint?
  • Subtract the exponents first.
  • Then rewrite as 1/xⁿ.
E3

Simplify z^2 ÷ z^3

Need a hint?
  • Subtract the exponents first.
  • Then rewrite as 1/xⁿ.
E4

Simplify x^1 ÷ x^4

Need a hint?
  • Subtract the exponents first.
  • Then rewrite as 1/xⁿ.