The quotient [[$ \dfrac{4^5}{4^2} $]] is called a quotient of powers with the same base.

Example 1

Simplify the powers.

a) [[$ \dfrac{2^6}{2^3} = 2^{6-3} = 2^3 = 8 $]]

b) [[$ \dfrac{(-4)^7}{(-4)^5} = (-4)^{7-5} = (-4)^2 = 16 $]]

c) [[$ \dfrac{y^7}{y^3} = y^{7-3} = y^4 $]]

d) [[$ \dfrac{a^3 \cdot a^4 \cdot a^6}{a^2 \cdot a^5} = \dfrac{a^{3+4+6}}{a^{2+5}} = \dfrac{a^{13}}{a^7} = a^{13-7} = a^6 $]]

There are often other factors involved in the quotient of powers with the same base, so you should be careful with the base number.

Example 2

Simplify the powers.

a) [[$ \dfrac{3x^4}{x} = 3x^{4-1} = 3x^3 $]]

b) [[$ \dfrac{-3^8}{3^5} = -3^{8-5} = -3^3 = -27 $]]

c) [[$ \dfrac{6a^4b^2}{3ab} = \dfrac{6}{3}a^{4-1}b^{2-1} = 2a^3b^1 = 2a^3b $]]

Divide the numbers and combine powers with same base.