Simplify ( cube root of 7y^2)/( cube root of 25x^2)

$\frac{\sqrt[3]{7 y^{2}}}{\sqrt[3]{25 x^{2}}}$

Step 1

Multiply $\frac{\sqrt[3]{7 y^{2}}}{\sqrt[3]{25 x^{2}}}$ by $\frac{{\sqrt[3]{25 x^{2}}}^{2}}{{\sqrt[3]{25 x^{2}}}^{2}}$ .

$\frac{\sqrt[3]{7 y^{2}}}{\sqrt[3]{25 x^{2}}} \cdot \frac{{\sqrt[3]{25 x^{2}}}^{2}}{{\sqrt[3]{25 x^{2}}}^{2}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{7 y^{2}}}{\sqrt[3]{25 x^{2}}}$ by $\frac{{\sqrt[3]{25 x^{2}}}^{2}}{{\sqrt[3]{25 x^{2}}}^{2}}$ .

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{25 x^{2}}}^{2}}{\sqrt[3]{25 x^{2}} {\sqrt[3]{25 x^{2}}}^{2}}$

Raise $\sqrt[3]{25 x^{2}}$ to the power of $1$ .

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{25 x^{2}}}^{2}}{{\sqrt[3]{25 x^{2}}}^{1} {\sqrt[3]{25 x^{2}}}^{2}}$

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

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{25 x^{2}}}^{2}}{{\sqrt[3]{25 x^{2}}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{25 x^{2}}}^{2}}{{\sqrt[3]{25 x^{2}}}^{3}}$

Rewrite ${\sqrt[3]{25 x^{2}}}^{3}$ as $25 x^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{25 x^{2}}$ as ${(25 x^{2})}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{25 x^{2}}}^{2}}{{({(25 x^{2})}^{\frac{1}{3}})}^{3}}$

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

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{25 x^{2}}}^{2}}{{(25 x^{2})}^{\frac{1}{3} \cdot 3}}$

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

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{25 x^{2}}}^{2}}{{(25 x^{2})}^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{25 x^{2}}}^{2}}{{(25 x^{2})}^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{25 x^{2}}}^{2}}{{(25 x^{2})}^{1}}$

Simplify.

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{25 x^{2}}}^{2}}{25 x^{2}}$

Step 3

Simplify the numerator.

Rewrite ${\sqrt[3]{25 x^{2}}}^{2}$ as $\sqrt[3]{{(25 x^{2})}^{2}}$ .

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{{(25 x^{2})}^{2}}}{25 x^{2}}$

Apply the product rule to $25 x^{2}$ .

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{25^{2} {(x^{2})}^{2}}}{25 x^{2}}$

Raise $25$ to the power of $2$ .

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{625 {(x^{2})}^{2}}}{25 x^{2}}$

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

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

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{625 x^{2 \cdot 2}}}{25 x^{2}}$

Multiply $2$ by $2$ .

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{625 x^{4}}}{25 x^{2}}$

Rewrite $625 x^{4}$ as ${(5 x)}^{3} \cdot (5 x)$ .

Factor $125$ out of $625$ .

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{125 (5) x^{4}}}{25 x^{2}}$

Rewrite $125$ as $5^{3}$ .

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{5^{3} \cdot 5 x^{4}}}{25 x^{2}}$

Factor out $x^{3}$ .

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{5^{3} \cdot 5 (x^{3} x)}}{25 x^{2}}$

Move $5$ .

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{5^{3} x^{3} \cdot 5 x}}{25 x^{2}}$

Rewrite $5^{3} x^{3}$ as ${(5 x)}^{3}$ .

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{{(5 x)}^{3} \cdot 5 x}}{25 x^{2}}$

Add parentheses.

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{{(5 x)}^{3} \cdot (5 x)}}{25 x^{2}}$

Pull terms out from under the radical.

$\frac{\sqrt[3]{7 y^{2}} \cdot 5 x \sqrt[3]{5 x}}{25 x^{2}}$

Combine exponents.

Combine using the product rule for radicals.

$\frac{5 x \sqrt[3]{7 y^{2} (5 x)}}{25 x^{2}}$

Multiply $5$ by $7$ .

$\frac{5 x \sqrt[3]{35 y^{2} x}}{25 x^{2}}$

Step 4

Reduce the expression by cancelling the common factors.

Cancel the common factor of $5$ and $25$ .

Factor $5$ out of $5 x \sqrt[3]{35 y^{2} x}$ .

$\frac{5 (x \sqrt[3]{35 y^{2} x})}{25 x^{2}}$

Cancel the common factors.

Factor $5$ out of $25 x^{2}$ .

$\frac{5 (x \sqrt[3]{35 y^{2} x})}{5 (5 x^{2})}$

Cancel the common factor.

$\frac{5 (x \sqrt[3]{35 y^{2} x})}{5 (5 x^{2})}$

Rewrite the expression.

$\frac{x \sqrt[3]{35 y^{2} x}}{5 x^{2}}$

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

Factor $x$ out of $x \sqrt[3]{35 y^{2} x}$ .

$\frac{x (\sqrt[3]{35 y^{2} x})}{5 x^{2}}$

Cancel the common factors.

Factor $x$ out of $5 x^{2}$ .

$\frac{x (\sqrt[3]{35 y^{2} x})}{x (5 x)}$

Cancel the common factor.

$\frac{x \sqrt[3]{35 y^{2} x}}{x (5 x)}$

Rewrite the expression.

$\frac{\sqrt[3]{35 y^{2} x}}{5 x}$

Simplify ( cube root of 7y^2)/( cube root of 25x^2)

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