Evaluate (9/4)^(-3/2)

${(\frac{9}{4})}^{- \frac{3}{2}}$

Step 1

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

${(\frac{4}{9})}^{\frac{3}{2}}$

Step 2

Apply the product rule to $\frac{4}{9}$ .

$\frac{4^{\frac{3}{2}}}{9^{\frac{3}{2}}}$

Step 3

Simplify the numerator.

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

$\frac{{(2^{2})}^{\frac{3}{2}}}{9^{\frac{3}{2}}}$

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

$\frac{2^{2 (\frac{3}{2})}}{9^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2^{2 (\frac{3}{2})}}{9^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{2^{3}}{9^{\frac{3}{2}}}$

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

$\frac{8}{9^{\frac{3}{2}}}$

Step 4

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{8}{{(3^{2})}^{\frac{3}{2}}}$

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

$\frac{8}{3^{2 (\frac{3}{2})}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{8}{3^{2 (\frac{3}{2})}}$

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:

$0.\overline{296}$