Simplify (81a^8b^2)^(1/4)

${(81 a^{8} b^{2})}^{\frac{1}{4}}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $81 a^{8} b^{2}$ .

${(81 a^{8})}^{\frac{1}{4}} {(b^{2})}^{\frac{1}{4}}$

Apply the product rule to $81 a^{8}$ .

$81^{\frac{1}{4}} {(a^{8})}^{\frac{1}{4}} {(b^{2})}^{\frac{1}{4}}$

Step 2

Simplify the expression.

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

${(3^{4})}^{\frac{1}{4}} {(a^{8})}^{\frac{1}{4}} {(b^{2})}^{\frac{1}{4}}$

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

$3^{4 (\frac{1}{4})} {(a^{8})}^{\frac{1}{4}} {(b^{2})}^{\frac{1}{4}}$

Step 3

Cancel the common factor of $4$ .

Cancel the common factor.

$3^{4 (\frac{1}{4})} {(a^{8})}^{\frac{1}{4}} {(b^{2})}^{\frac{1}{4}}$

Rewrite the expression.

$3^{1} {(a^{8})}^{\frac{1}{4}} {(b^{2})}^{\frac{1}{4}}$

Step 4

Evaluate the exponent.

$3 {(a^{8})}^{\frac{1}{4}} {(b^{2})}^{\frac{1}{4}}$

Step 5

Multiply the exponents in ${(a^{8})}^{\frac{1}{4}}$ .

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

$3 a^{8 (\frac{1}{4})} {(b^{2})}^{\frac{1}{4}}$

Cancel the common factor of $4$ .

Factor $4$ out of $8$ .

$3 a^{4 (2) \frac{1}{4}} {(b^{2})}^{\frac{1}{4}}$

Cancel the common factor.

$3 a^{4 \cdot 2 \frac{1}{4}} {(b^{2})}^{\frac{1}{4}}$

Rewrite the expression.

$3 a^{2} {(b^{2})}^{\frac{1}{4}}$

Step 6

Multiply the exponents in ${(b^{2})}^{\frac{1}{4}}$ .

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

$3 a^{2} b^{2 (\frac{1}{4})}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$3 a^{2} b^{2 \frac{1}{2 (2)}}$

Cancel the common factor.

$3 a^{2} b^{2 \frac{1}{2 \cdot 2}}$

Rewrite the expression.

$3 a^{2} b^{\frac{1}{2}}$