What is the value of (81)^-1/4?

To evaluate ( (81)^{-1/4} ), start by rewriting the expression using exponent rules. The expression ( a^{-n} ) can be rewritten as ( \frac{1}{a^n} ), so we can express ( (81)^{-1/4} ) as ( \frac{1}{(81)^{1/4}} ).

Next, calculate ( (81)^{1/4} ). Since ( 81 = 3^4 ), we can determine the fourth root of ( 81 ). Thus,

[

(81)^{1/4} = (3^4)^{1/4} = 3^{(4 \times \frac{1}{4})} = 3^1 = 3.

]

Now, substituting this back into our expression gives us

[

(81)^{-1/4} = \frac{1}{(81)^{1/4}} = \frac{1}{3}.

]

This reveals that the value of ( (81)^{-1/4} ) is ( \frac{1}{3} ), confirming that the choice is indeed accurate.

Understanding