Simplify this

Question

(2/5)^-3

(Simplify your answer. Type a fraction or mixed number.)

Answer 1

     (2/5)^-3

Recall that the quotient to a power property of exponents states that to raise a quotient to a power, raise the numerator and the denominator to that power.

For instance,          (a/b)ⁿ = aⁿ/bⁿ

Therefore,

               (2/5)^-3 = 2^-3/5^-3 

Negative exponents indicate reciprocation, so to make a negative exponent positive we take its reciprocal.

For instance,       a^-n = 1/aⁿ          and           1/a^-n = aⁿ

Therefore, 

                 2^-3/5^-3 = (2^-3)·(1/5^-3) = (1/2³)·(5³) = 5³/2³ 

Thus,

          5³/2³ = 125/8

          ==>   125/8 = 120/8 + 5/8 = 15 + 5/8 = 15 ⁵/₈

In summary,

          (2/5)^-3 = 2^-3/5^-3 = 5³/2³ = 125/8 = 15 ⁵/₈