Question

Explain why 81 3/4 equals 27

Answer 1

81 to the 4th root is 3 and 3 cubed is 27.

Answer 2

General Idea:

There are few exponent rules that need to be used to simplify the rational exponent to get to the answer.

Exponent rules:

`Rule\; 1: a \cdot a \cdot a \cdot a =a^4\\ \\ Rule\; 2: (a^m)^n=a^(m * n)`

Applying the concept:

Rewriting 81 as a number with an exponent using prime factorization.

`81=3 * 3 * 3 * 3 = 3^4\\ \\ Applying \; Rule \; 1, \; we\; get\; \\\\81^{(3)/(4)}=(3^4)^{(3)/(4)} \\ \\ Applying \; Rule \; 2,\; we\; get\\ \\ 3^{4 *(3)/(4) } = 3^{(12)/(4) }=3^3=3 * 3 * 3 = 27`

Conclusion:

The above explanation explains why
`81^{(3)/(4)} =27`