Powers · Divide — quotient rule · 4.4

When the bottom wins.

If the bottom exponent is bigger, subtracting gives a negative exponent — which means a leftover factor in the DENOMINATOR. x³ ÷ x⁵ = x⁻² = 1/x². This bridges to negative exponents.

01 / THE IDEA

See it work.

Watch the method

When the bottom wins

x³ ÷ x⁵ = x⁻² = 1/x².

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x3x5\frac{x^{3}}{x^{5}}
1

Spot the negative exponent

A negative exponent belongs in the denominator.

x3x5\frac{x^{3}}{x^{5}}
2

Rewrite with positive exponents

Flip the negative-exponent factors to the bottom.

=1x2= \frac{1}{x^{2}}
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.

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— / WHY IT WORKS

The idea underneath.

Leftover on the bottom

Cancel what you can; the extra factors that remain are in the denominator, which is exactly a negative exponent.

Why it works

xxx ÷ xxxxx = 1/xx.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify x^2 ÷ x^4

1

Spot the negative exponent

A negative exponent belongs in the denominator. \fracx^2x^4

2

Rewrite with positive exponents

Flip the negative-exponent factors to the bottom. = 1/x^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 x^2 ÷ x^4

1

Spot the negative exponent

A negative exponent belongs in the denominator. \fracx^2x^4

2

Rewrite with positive exponents

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

04 / COMMON TRAPS

Where students slip.

Trap

Writing a negative answer (−x²).

Fix

A negative EXPONENT means 1/x², not a negative number.

Trap

Subtracting the wrong way.

Fix

Top minus bottom; if negative, it goes to the denominator.

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 y^1 ÷ y^3

Need a hint?
  • Subtract top minus bottom.
  • Negative exponent → put it in the denominator.
C2

Simplify x^1 ÷ x^4

Need a hint?
  • Subtract top minus bottom.
  • Negative exponent → put it in the denominator.
C3

Simplify y^2 ÷ y^5

Need a hint?
  • Subtract top minus bottom.
  • Negative exponent → put it in the denominator.
C4

Simplify z^1 ÷ z^2

Need a hint?
  • Subtract top minus bottom.
  • Negative exponent → put it in the denominator.
Apply

Apply (4)

A1

Simplify x^2 ÷ x^3

Need a hint?
  • Subtract top minus bottom.
  • Negative exponent → put it in the denominator.
A2

Simplify z^2 ÷ z^4

Need a hint?
  • Subtract top minus bottom.
  • Negative exponent → put it in the denominator.
A3

Simplify x^2 ÷ x^5

Need a hint?
  • Subtract top minus bottom.
  • Negative exponent → put it in the denominator.
A4

Simplify y^2 ÷ y^3

Need a hint?
  • Subtract top minus bottom.
  • Negative exponent → put it in the denominator.
Extension

Extension (4)

E1

Simplify x^1 ÷ x^3

Need a hint?
  • Subtract top minus bottom.
  • Negative exponent → put it in the denominator.
E2

Simplify z^2 ÷ z^4

Need a hint?
  • Subtract top minus bottom.
  • Negative exponent → put it in the denominator.
E3

Simplify y^1 ÷ y^2

Need a hint?
  • Subtract top minus bottom.
  • Negative exponent → put it in the denominator.
E4

Simplify y^2 ÷ y^3

Need a hint?
  • Subtract top minus bottom.
  • Negative exponent → put it in the denominator.