# Quotient of Powers

How to solve the quotient of powers a^{m}/a^{n}: formula, 2 examples, and their solutions.

## Formula

a^{m}/a^{n} = a^{m - n}

(a ≠ 0)

When dividing the same base powers,

subtract the exponents: m - n.

Product of Powers

## Examplex^{8}/x^{3}

Solution

Solution (Detail)

x^{8} and x^{3} have the same base: x.

So x^{8}/x^{3} = x^{8 - 3}.

8 - 3 = 5.

So x^{5} is the answer.

## Example[12x^{7}y^{2}z^{9}]/[4x^{4}y^{2}z^{5}]

Solution

Solution (Detail)

First simplify the coefficient numbers.

(= numbers in front of x, y, z)

12/4 = 3

So Write 3.

x^{7} and x^{4} have the same base: x.

So x^{7}/x^{4} = x^{7 - 4}.

Cancel the y^{2}

on both of the numerator and the denominator.

So write 1.

z^{9} and z^{5} have the same base: z.

So z^{9}/z^{5} = z^{9 - 5}.

So (given) = 3⋅x^{7 - 4}⋅1⋅z^{9 - 5}.

The same base powers are combined.

7 - 4 = 3

9 - 5 = 4

So 3x^{3}z^{4} is the answer.