Simplify ( fourth root of 81x^8y^12)^3

${(\sqrt[4]{81 x^{8} y^{12}})}^{3}$

Step 1

Rewrite $81 x^{8} y^{12}$ as ${(3 x^{2} y^{3})}^{4}$ .

${\sqrt[4]{{(3 x^{2} y^{3})}^{4}}}^{3}$

Step 2

Pull terms out from under the radical, assuming positive real numbers.

${(3 x^{2} y^{3})}^{3}$

Step 3

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

Apply the product rule to $3 x^{2} y^{3}$ .

${(3 x^{2})}^{3} {(y^{3})}^{3}$

Apply the product rule to $3 x^{2}$ .

$3^{3} {(x^{2})}^{3} {(y^{3})}^{3}$

Step 4

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

$27 {(x^{2})}^{3} {(y^{3})}^{3}$

Step 5

Multiply the exponents in ${(x^{2})}^{3}$ .

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

$27 x^{2 \cdot 3} {(y^{3})}^{3}$

Multiply $2$ by $3$ .

$27 x^{6} {(y^{3})}^{3}$

Step 6

Multiply the exponents in ${(y^{3})}^{3}$ .

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

$27 x^{6} y^{3 \cdot 3}$

Multiply $3$ by $3$ .

$27 x^{6} y^{9}$