Dividing Powers and the Zero Exponent

≈ 25 min

Dividing powers with the same base works in reverse — you subtract the exponents:

a^m \div a^n = a^{m-n} \quad (a \neq 0)

This is because dividing cancels matching factors top and bottom:

\frac{2^5}{2^2} = \frac{2 \times 2 \times 2 \times 2 \times 2}{2 \times 2} = 2^3

Two of the twos cancel, leaving $5 - 2 = 3$ of them.

Worked example. Simplify $\dfrac{y^7}{y^4}$ .

Same base, subtract the exponents: $y^{7-4} = y^3$ .

What is $7^0$ ?

$0$

$1$

$7$

$70$

Evaluate $2^5 \div 2^2$ .

Write $5^6 \div 5^2$ as a single power, in the form base^exponent.

Simplify $\dfrac{x^8}{x^3}$ . Write your answer in the form x^n.