Powers · Divide — quotient rule · 4.1

Quotient rule.

To divide powers of the SAME base, keep the base and SUBTRACT the exponents: xᵃ ÷ xᵇ = xᵃ⁻ᵇ. It works because common factors cancel. x⁷ ÷ x⁴ = x³.

01 / THE IDEA

See it work.

See the factors

Cancel matching factors.

Every power is just repeated multiplication.

xxxxxxx
xxxx

Watch the method

Quotient rule

x⁷ ÷ x⁴ = x³ — cancel four x’s.

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x7x4\frac{x^{7}}{x^{4}}
1

Subtract the exponents

Same base x — keep the base, subtract: 7 − 4.

x7÷x4=x3x^{7} \div x^{4} = x^{3}
2

Write the quotient

Same base when dividing → subtract exponents.

x7x4=x3\frac{x^{7}}{x^{4}} = x^{3}
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.

Cancel common factors

Four of the seven x’s on top cancel the four on the bottom, leaving x³. Hence subtract the exponents.

Why it works

xxxxxxx ÷ xxxx = xxx.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify y^5 ÷ y^3

1

Subtract the exponents

Same base y — keep the base, subtract: 5 − 3. y^5 ÷ y^3 = y^2

2

Write the quotient

Same base when dividing → subtract exponents. \fracy^5y^3 = y^2

EX 2medium

Simplify z^7 ÷ z^3

1

Subtract the exponents

Same base z — keep the base, subtract: 7 − 3. z^7 ÷ z^3 = z^4

2

Write the quotient

Same base when dividing → subtract exponents. \fracz^7z^3 = z^4

EX 3hard

Simplify x^5 ÷ x^2

1

Subtract the exponents

Same base x — keep the base, subtract: 5 − 2. x^5 ÷ x^2 = x^3

2

Write the quotient

Same base when dividing → subtract exponents. \fracx^5x^2 = x^3

04 / COMMON TRAPS

Where students slip.

Trap

Dividing the exponents.

Fix

Dividing powers SUBTRACTS exponents, not divides them.

Trap

Subtracting the wrong way.

Fix

Top exponent minus bottom exponent.

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^6 ÷ y^2

Need a hint?
  • Keep the base; subtract the exponents.
  • Cancel to check.
C2

Simplify y^7 ÷ y^3

Need a hint?
  • Keep the base; subtract the exponents.
  • Cancel to check.
C3

Simplify y^5 ÷ y^2

Need a hint?
  • Keep the base; subtract the exponents.
  • Cancel to check.
C4

Simplify y^6 ÷ y^3

Need a hint?
  • Keep the base; subtract the exponents.
  • Cancel to check.
Apply

Apply (4)

A1

Simplify y^4 ÷ y^2

Need a hint?
  • Keep the base; subtract the exponents.
  • Cancel to check.
A2

Simplify y^5 ÷ y^3

Need a hint?
  • Keep the base; subtract the exponents.
  • Cancel to check.
A3

Simplify z^4 ÷ z^2

Need a hint?
  • Keep the base; subtract the exponents.
  • Cancel to check.
A4

Simplify z^6 ÷ z^3

Need a hint?
  • Keep the base; subtract the exponents.
  • Cancel to check.
Extension

Extension (4)

E1

Simplify y^4 ÷ y^3

Need a hint?
  • Keep the base; subtract the exponents.
  • Cancel to check.
E2

Simplify z^5 ÷ z^3

Need a hint?
  • Keep the base; subtract the exponents.
  • Cancel to check.
E3

Simplify x^5 ÷ x^2

Need a hint?
  • Keep the base; subtract the exponents.
  • Cancel to check.
E4

Simplify x^6 ÷ x^3

Need a hint?
  • Keep the base; subtract the exponents.
  • Cancel to check.