Simplify ((8x^(1/4))/(y^9))^(2/3)

${(\frac{8 x^{\frac{1}{4}}}{y^{9}})}^{\frac{2}{3}}$

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

Apply the product rule to $\frac{8 x^{\frac{1}{4}}}{y^{9}}$ .

$\frac{{(8 x^{\frac{1}{4}})}^{\frac{2}{3}}}{{(y^{9})}^{\frac{2}{3}}}$

Apply the product rule to $8 x^{\frac{1}{4}}$ .

$\frac{8^{\frac{2}{3}} {(x^{\frac{1}{4}})}^{\frac{2}{3}}}{{(y^{9})}^{\frac{2}{3}}}$

Simplify the numerator.

Rewrite $8$ as $2^{3}$ .

$\frac{{(2^{3})}^{\frac{2}{3}} {(x^{\frac{1}{4}})}^{\frac{2}{3}}}{{(y^{9})}^{\frac{2}{3}}}$

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

$\frac{2^{3 (\frac{2}{3})} {(x^{\frac{1}{4}})}^{\frac{2}{3}}}{{(y^{9})}^{\frac{2}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{2^{3 (\frac{2}{3})} {(x^{\frac{1}{4}})}^{\frac{2}{3}}}{{(y^{9})}^{\frac{2}{3}}}$

Rewrite the expression.

$\frac{2^{2} {(x^{\frac{1}{4}})}^{\frac{2}{3}}}{{(y^{9})}^{\frac{2}{3}}}$

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

$\frac{4 {(x^{\frac{1}{4}})}^{\frac{2}{3}}}{{(y^{9})}^{\frac{2}{3}}}$

Multiply the exponents in ${(x^{\frac{1}{4}})}^{\frac{2}{3}}$ .

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

$\frac{4 x^{\frac{1}{4} \cdot \frac{2}{3}}}{{(y^{9})}^{\frac{2}{3}}}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{4 x^{\frac{1}{2 (2)} \cdot \frac{2}{3}}}{{(y^{9})}^{\frac{2}{3}}}$

Cancel the common factor.

$\frac{4 x^{\frac{1}{2 \cdot 2} \cdot \frac{2}{3}}}{{(y^{9})}^{\frac{2}{3}}}$

Rewrite the expression.

$\frac{4 x^{\frac{1}{2} \cdot \frac{1}{3}}}{{(y^{9})}^{\frac{2}{3}}}$

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

$\frac{4 x^{\frac{1}{2 \cdot 3}}}{{(y^{9})}^{\frac{2}{3}}}$

Multiply $2$ by $3$ .

$\frac{4 x^{\frac{1}{6}}}{{(y^{9})}^{\frac{2}{3}}}$

Multiply the exponents in ${(y^{9})}^{\frac{2}{3}}$ .

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

$\frac{4 x^{\frac{1}{6}}}{y^{9 (\frac{2}{3})}}$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$\frac{4 x^{\frac{1}{6}}}{y^{3 (3) \frac{2}{3}}}$

Cancel the common factor.

$\frac{4 x^{\frac{1}{6}}}{y^{3 \cdot 3 \frac{2}{3}}}$

Rewrite the expression.

$\frac{4 x^{\frac{1}{6}}}{y^{3 \cdot 2}}$

Multiply $3$ by $2$ .

$\frac{4 x^{\frac{1}{6}}}{y^{6}}$