Simplify ((4x^(2/5))/(9y^(4/5)))^(5/2)

${(\frac{4 x^{\frac{2}{5}}}{9 y^{\frac{4}{5}}})}^{\frac{5}{2}}$

Step 1

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

Apply the product rule to $\frac{4 x^{\frac{2}{5}}}{9 y^{\frac{4}{5}}}$ .

$\frac{{(4 x^{\frac{2}{5}})}^{\frac{5}{2}}}{{(9 y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Apply the product rule to $4 x^{\frac{2}{5}}$ .

$\frac{4^{\frac{5}{2}} {(x^{\frac{2}{5}})}^{\frac{5}{2}}}{{(9 y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Apply the product rule to $9 y^{\frac{4}{5}}$ .

$\frac{4^{\frac{5}{2}} {(x^{\frac{2}{5}})}^{\frac{5}{2}}}{9^{\frac{5}{2}} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Step 2

Simplify the numerator.

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

$\frac{{(2^{2})}^{\frac{5}{2}} {(x^{\frac{2}{5}})}^{\frac{5}{2}}}{9^{\frac{5}{2}} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

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

$\frac{2^{2 (\frac{5}{2})} {(x^{\frac{2}{5}})}^{\frac{5}{2}}}{9^{\frac{5}{2}} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2^{2 (\frac{5}{2})} {(x^{\frac{2}{5}})}^{\frac{5}{2}}}{9^{\frac{5}{2}} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Rewrite the expression.

$\frac{2^{5} {(x^{\frac{2}{5}})}^{\frac{5}{2}}}{9^{\frac{5}{2}} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Raise $2$ to the power of $5$ .

$\frac{32 {(x^{\frac{2}{5}})}^{\frac{5}{2}}}{9^{\frac{5}{2}} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Multiply the exponents in ${(x^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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

$\frac{32 x^{\frac{2}{5} \cdot \frac{5}{2}}}{9^{\frac{5}{2}} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{32 x^{\frac{2}{5} \cdot \frac{5}{2}}}{9^{\frac{5}{2}} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Rewrite the expression.

$\frac{32 x^{\frac{1}{5} \cdot 5}}{9^{\frac{5}{2}} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{32 x^{\frac{1}{5} \cdot 5}}{9^{\frac{5}{2}} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Rewrite the expression.

$\frac{32 x^{1}}{9^{\frac{5}{2}} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Simplify.

$\frac{32 x}{9^{\frac{5}{2}} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Step 3

Simplify the denominator.

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

$\frac{32 x}{{(3^{2})}^{\frac{5}{2}} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

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

$\frac{32 x}{3^{2 (\frac{5}{2})} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{32 x}{3^{2 (\frac{5}{2})} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Rewrite the expression.

$\frac{32 x}{3^{5} {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Raise $3$ to the power of $5$ .

$\frac{32 x}{243 {(y^{\frac{4}{5}})}^{\frac{5}{2}}}$

Multiply the exponents in ${(y^{\frac{4}{5}})}^{\frac{5}{2}}$ .

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

$\frac{32 x}{243 y^{\frac{4}{5} \cdot \frac{5}{2}}}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{32 x}{243 y^{\frac{2 (2)}{5} \cdot \frac{5}{2}}}$

Cancel the common factor.

$\frac{32 x}{243 y^{\frac{2 \cdot 2}{5} \cdot \frac{5}{2}}}$

Rewrite the expression.

$\frac{32 x}{243 y^{\frac{2}{5} \cdot 5}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{32 x}{243 y^{\frac{2}{5} \cdot 5}}$

Rewrite the expression.

$\frac{32 x}{243 y^{2}}$

Simplify ((4x^(2/5))/(9y^(4/5)))^(5/2)

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