General Idea:
There are few exponent rules that need to be used to simplify the rational exponent to get to the answer.
Exponent rules:
[tex] Rule\; 1: a \cdot a \cdot a \cdot a =a^4\\ \\ Rule\; 2: (a^m)^n=a^{m \times n} [/tex]
Applying the concept:
Rewriting 81 as a number with an exponent using prime factorization.
[tex] 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 [/tex]
Conclusion:
The above explanation explains why [tex] 81^{\frac{3}{4}} =27 [/tex]
