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:

$Rule\; 1: a \cdot a \cdot a \cdot a =a^4\\ \\ Rule\; 2: (a^m)^n=a^{m \times n}$

Applying the concept:

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

$81=3 \times 3 \times 3 \times 3 = 3^4\\ \\ Applying \; Rule \; 1, \; we\; get\; \\\\81^{\frac{3}{4}}=(3^4)^{\frac{3}{4}} \\ \\ Applying \; Rule \; 2,\; we\; get\\ \\ 3^{4 \times\frac{3}{4} } = 3^{\frac{12}{4} }=3^3=3 \times 3 \times 3 = 27$

Conclusion:

The above explanation explains why $81^{\frac{3}{4}} =27$