Division Rule

Q: What if the exponents are equal?

Then everything cancels: $\dfrac{x^{4}}{x^{4}} = x^{0} = 1$. Any nonzero base to the zero power is $1$, and this is exactly where that fact comes from.

When you divide two powers with the same base, you keep the base and subtract the exponents: $\dfrac{x^{9}}{x^{4}} = x^{5}$ . That's the division rule (also called the quotient rule), and it's the mirror image of the multiplication rule.

Here's why it works: $\dfrac{x^{9}}{x^{4}}$ is nine $x$ 's on top and four on the bottom. Each $x$ on the bottom cancels one on top, wiping out four pairs and leaving five $x$ 's upstairs — $x^{5}$ . Subtracting the exponents is just fast cancelling.

The rule

Same base, divided: keep the base, subtract the exponents — top minus bottom. In symbols, $\dfrac{x^{m}}{x^{n}} = x^{m-n}$ .

Just like with multiplication, the base must match, and a plain variable carries an invisible exponent of $1$ . So $\dfrac{a^{10}}{a} = a^{10-1} = a^{9}$ .

Coefficients divide separately

The numbers in front follow normal division, and only the exponents get subtracted. In $\dfrac{18y^{7}}{6y^{3}}$ , divide $18 \div 6 = 3$ and subtract $7 - 3 = 4$ , giving $3y^{4}$ .

With more than one variable, subtract each variable's exponents on its own: $\dfrac{20a^{5}b^{8}}{4a^{2}b^{3}}$ gives coefficient $5$ , then $a^{5-2} = a^{3}$ and $b^{8-3} = b^{5}$ , so the answer is $5a^{3}b^{5}$ .

Order matters in the subtraction

Always subtract the bottom exponent from the top exponent. If the bigger exponent is on the bottom, the subtraction comes out negative — $\dfrac{x^{3}}{x^{7}} = x^{-4}$ — and a negative exponent means the power really lives in the denominator: $\dfrac{1}{x^{4}}$ . That's the negative exponent rule, which gets its own lesson.

Worked examples

Example 1: same base, subtract the exponents

Simplify $\dfrac{x^{9}}{x^{4}}$ .

Same base

x

, so keep it and subtract the exponents

x^{9-4}

Subtract

x^{5}

Answer: $x^{5}$

Example 2: coefficients divide, exponents subtract

Simplify $\dfrac{24x^{6}}{8x^{2}}$ .

Divide the coefficients

24 \div 8 = 3

Subtract the exponents on

x

6 - 2 = 4

Put them together

3x^{4}

Answer: $3x^{4}$

Example 3: two variables

Simplify $\dfrac{20a^{5}b^{8}}{4a^{2}b^{3}}$ .

Divide the coefficients

20 \div 4 = 5

Subtract the exponents on

a

5 - 2 = 3

Subtract the exponents on

b

8 - 3 = 5

Put them together

5a^{3}b^{5}

Answer: $5a^{3}b^{5}$

Try one yourself

Common questions

Why subtract the exponents instead of dividing them?

Because every factor on the bottom cancels one matching factor on top. Nine $x$ 's over four $x$ 's leaves $9 - 4 = 5$ of them. Subtraction counts what survives the cancelling.

What if the exponents are equal?

Then everything cancels: $\dfrac{x^{4}}{x^{4}} = x^{0} = 1$ . Any nonzero base to the zero power is $1$ , and this is exactly where that fact comes from.

What if the bottom exponent is bigger?

The subtraction gives a negative exponent: $\dfrac{x^{2}}{x^{5}} = x^{-3}$ . Rewrite it with a positive exponent as $\dfrac{1}{x^{3}}$ — the leftover factors are on the bottom.

Do I subtract the coefficients too?

No — coefficients divide normally. In $\dfrac{12x^{5}}{3x^{2}}$ , the $12$ and $3$ divide to $4$ while the exponents subtract to $3$ , giving $4x^{3}$ .

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Division Rule

The rule

Coefficients divide separately

Order matters in the subtraction

Worked examples

Example 1: same base, subtract the exponents

Example 2: coefficients divide, exponents subtract

Example 3: two variables

Try one yourself

Common questions

Keep learning

Multiplication Rule

Power Rule

Negative Exponent Rule

Dividing Polynomials