Evaluate (81/16)^(-3/4)

${(\frac{81}{16})}^{- \frac{3}{4}}$

Step 1

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{16}{81})}^{\frac{3}{4}}$

Step 2

Apply the product rule to $\frac{16}{81}$ .

$\frac{16^{\frac{3}{4}}}{81^{\frac{3}{4}}}$

Step 3

Simplify the numerator.

Rewrite $16$ as $2^{4}$ .

$\frac{{(2^{4})}^{\frac{3}{4}}}{81^{\frac{3}{4}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2^{4 (\frac{3}{4})}}{81^{\frac{3}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{2^{4 (\frac{3}{4})}}{81^{\frac{3}{4}}}$

Rewrite the expression.

$\frac{2^{3}}{81^{\frac{3}{4}}}$

Raise $2$ to the power of $3$ .

$\frac{8}{81^{\frac{3}{4}}}$

Step 4

Simplify the denominator.

Rewrite $81$ as $3^{4}$ .

$\frac{8}{{(3^{4})}^{\frac{3}{4}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{8}{3^{4 (\frac{3}{4})}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{8}{3^{4 (\frac{3}{4})}}$

Rewrite the expression.

$\frac{8}{3^{3}}$

Raise $3$ to the power of $3$ .

$\frac{8}{27}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{8}{27}$

Decimal Form:

$0.\overline{296}$