Evaluate (1/81)^(1/2)*(1/81)^(-3/4)

${(\frac{1}{81})}^{\frac{1}{2}} \cdot {(\frac{1}{81})}^{- \frac{3}{4}}$

Step 1

Multiply ${(\frac{1}{81})}^{\frac{1}{2}}$ by ${(\frac{1}{81})}^{- \frac{3}{4}}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{1}{81})}^{\frac{1}{2} - \frac{3}{4}}$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(\frac{1}{81})}^{\frac{1}{2} \cdot \frac{2}{2} - \frac{3}{4}}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

${(\frac{1}{81})}^{\frac{2}{2 \cdot 2} - \frac{3}{4}}$

Multiply $2$ by $2$ .

${(\frac{1}{81})}^{\frac{2}{4} - \frac{3}{4}}$

Combine the numerators over the common denominator.

${(\frac{1}{81})}^{\frac{2 - 3}{4}}$

Subtract $3$ from $2$ .

${(\frac{1}{81})}^{\frac{- 1}{4}}$

Move the negative in front of the fraction.

${(\frac{1}{81})}^{- \frac{1}{4}}$

Step 2

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

$81^{\frac{1}{4}}$

Step 3

Simplify the expression.

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

${(3^{4})}^{\frac{1}{4}}$

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

$3^{4 (\frac{1}{4})}$

Step 4

Cancel the common factor of $4$ .

Cancel the common factor.

$3^{4 (\frac{1}{4})}$

Rewrite the expression.

$3^{1}$

Step 5

Evaluate the exponent.

$3$