Simplify ( cube root of 5y^2)/( cube root of 4x^2)

$\frac{\sqrt[3]{5 y^{2}}}{\sqrt[3]{4 x^{2}}}$

Step 1

Multiply $\frac{\sqrt[3]{5 y^{2}}}{\sqrt[3]{4 x^{2}}}$ by $\frac{{\sqrt[3]{4 x^{2}}}^{2}}{{\sqrt[3]{4 x^{2}}}^{2}}$ .

$\frac{\sqrt[3]{5 y^{2}}}{\sqrt[3]{4 x^{2}}} \cdot \frac{{\sqrt[3]{4 x^{2}}}^{2}}{{\sqrt[3]{4 x^{2}}}^{2}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{5 y^{2}}}{\sqrt[3]{4 x^{2}}}$ by $\frac{{\sqrt[3]{4 x^{2}}}^{2}}{{\sqrt[3]{4 x^{2}}}^{2}}$ .

$\frac{\sqrt[3]{5 y^{2}} {\sqrt[3]{4 x^{2}}}^{2}}{\sqrt[3]{4 x^{2}} {\sqrt[3]{4 x^{2}}}^{2}}$

Raise $\sqrt[3]{4 x^{2}}$ to the power of $1$ .

$\frac{\sqrt[3]{5 y^{2}} {\sqrt[3]{4 x^{2}}}^{2}}{{\sqrt[3]{4 x^{2}}}^{1} {\sqrt[3]{4 x^{2}}}^{2}}$

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

$\frac{\sqrt[3]{5 y^{2}} {\sqrt[3]{4 x^{2}}}^{2}}{{\sqrt[3]{4 x^{2}}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{\sqrt[3]{5 y^{2}} {\sqrt[3]{4 x^{2}}}^{2}}{{\sqrt[3]{4 x^{2}}}^{3}}$

Rewrite ${\sqrt[3]{4 x^{2}}}^{3}$ as $4 x^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x^{2}}$ as ${(4 x^{2})}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{5 y^{2}} {\sqrt[3]{4 x^{2}}}^{2}}{{({(4 x^{2})}^{\frac{1}{3}})}^{3}}$

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

$\frac{\sqrt[3]{5 y^{2}} {\sqrt[3]{4 x^{2}}}^{2}}{{(4 x^{2})}^{\frac{1}{3} \cdot 3}}$

Combine $\frac{1}{3}$ and $3$ .

$\frac{\sqrt[3]{5 y^{2}} {\sqrt[3]{4 x^{2}}}^{2}}{{(4 x^{2})}^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{\sqrt[3]{5 y^{2}} {\sqrt[3]{4 x^{2}}}^{2}}{{(4 x^{2})}^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{\sqrt[3]{5 y^{2}} {\sqrt[3]{4 x^{2}}}^{2}}{{(4 x^{2})}^{1}}$

Simplify.

$\frac{\sqrt[3]{5 y^{2}} {\sqrt[3]{4 x^{2}}}^{2}}{4 x^{2}}$

Step 3

Simplify the numerator.

Rewrite ${\sqrt[3]{4 x^{2}}}^{2}$ as $\sqrt[3]{{(4 x^{2})}^{2}}$ .

$\frac{\sqrt[3]{5 y^{2}} \sqrt[3]{{(4 x^{2})}^{2}}}{4 x^{2}}$

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

$\frac{\sqrt[3]{5 y^{2}} \sqrt[3]{4^{2} {(x^{2})}^{2}}}{4 x^{2}}$

Raise $4$ to the power of $2$ .

$\frac{\sqrt[3]{5 y^{2}} \sqrt[3]{16 {(x^{2})}^{2}}}{4 x^{2}}$

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

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

$\frac{\sqrt[3]{5 y^{2}} \sqrt[3]{16 x^{2 \cdot 2}}}{4 x^{2}}$

Multiply $2$ by $2$ .

$\frac{\sqrt[3]{5 y^{2}} \sqrt[3]{16 x^{4}}}{4 x^{2}}$

Rewrite $16 x^{4}$ as ${(2 x)}^{3} \cdot (2 x)$ .

Factor $8$ out of $16$ .

$\frac{\sqrt[3]{5 y^{2}} \sqrt[3]{8 (2) x^{4}}}{4 x^{2}}$

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

$\frac{\sqrt[3]{5 y^{2}} \sqrt[3]{2^{3} \cdot 2 x^{4}}}{4 x^{2}}$

Factor out $x^{3}$ .

$\frac{\sqrt[3]{5 y^{2}} \sqrt[3]{2^{3} \cdot 2 (x^{3} x)}}{4 x^{2}}$

Move $2$ .

$\frac{\sqrt[3]{5 y^{2}} \sqrt[3]{2^{3} x^{3} \cdot 2 x}}{4 x^{2}}$

Rewrite $2^{3} x^{3}$ as ${(2 x)}^{3}$ .

$\frac{\sqrt[3]{5 y^{2}} \sqrt[3]{{(2 x)}^{3} \cdot 2 x}}{4 x^{2}}$

Add parentheses.

$\frac{\sqrt[3]{5 y^{2}} \sqrt[3]{{(2 x)}^{3} \cdot (2 x)}}{4 x^{2}}$

Pull terms out from under the radical.

$\frac{\sqrt[3]{5 y^{2}} \cdot 2 x \sqrt[3]{2 x}}{4 x^{2}}$

Combine exponents.

Combine using the product rule for radicals.

$\frac{2 x \sqrt[3]{5 y^{2} (2 x)}}{4 x^{2}}$

Multiply $2$ by $5$ .

$\frac{2 x \sqrt[3]{10 y^{2} x}}{4 x^{2}}$

Step 4

Reduce the expression by cancelling the common factors.

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 x \sqrt[3]{10 y^{2} x}$ .

$\frac{2 (x \sqrt[3]{10 y^{2} x})}{4 x^{2}}$

Cancel the common factors.

Factor $2$ out of $4 x^{2}$ .

$\frac{2 (x \sqrt[3]{10 y^{2} x})}{2 (2 x^{2})}$

Cancel the common factor.

$\frac{2 (x \sqrt[3]{10 y^{2} x})}{2 (2 x^{2})}$

Rewrite the expression.

$\frac{x \sqrt[3]{10 y^{2} x}}{2 x^{2}}$

Cancel the common factor of $x$ and $x^{2}$ .

Factor $x$ out of $x \sqrt[3]{10 y^{2} x}$ .

$\frac{x (\sqrt[3]{10 y^{2} x})}{2 x^{2}}$

Cancel the common factors.

Factor $x$ out of $2 x^{2}$ .

$\frac{x (\sqrt[3]{10 y^{2} x})}{x (2 x)}$

Cancel the common factor.

$\frac{x \sqrt[3]{10 y^{2} x}}{x (2 x)}$

Rewrite the expression.

$\frac{\sqrt[3]{10 y^{2} x}}{2 x}$

Simplify ( cube root of 5y^2)/( cube root of 4x^2)

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