Simplify cube root of (7x^2)/(15yz^2)

$\sqrt[3]{\frac{7 x^{2}}{15 y z^{2}}}$

Step 1

Rewrite $\sqrt[3]{\frac{7 x^{2}}{15 y z^{2}}}$ as $\frac{\sqrt[3]{7 x^{2}}}{\sqrt[3]{15 y z^{2}}}$ .

$\frac{\sqrt[3]{7 x^{2}}}{\sqrt[3]{15 y z^{2}}}$

Step 2

Multiply $\frac{\sqrt[3]{7 x^{2}}}{\sqrt[3]{15 y z^{2}}}$ by $\frac{{\sqrt[3]{15 y z^{2}}}^{2}}{{\sqrt[3]{15 y z^{2}}}^{2}}$ .

$\frac{\sqrt[3]{7 x^{2}}}{\sqrt[3]{15 y z^{2}}} \cdot \frac{{\sqrt[3]{15 y z^{2}}}^{2}}{{\sqrt[3]{15 y z^{2}}}^{2}}$

Step 3

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{7 x^{2}}}{\sqrt[3]{15 y z^{2}}}$ by $\frac{{\sqrt[3]{15 y z^{2}}}^{2}}{{\sqrt[3]{15 y z^{2}}}^{2}}$ .

$\frac{\sqrt[3]{7 x^{2}} {\sqrt[3]{15 y z^{2}}}^{2}}{\sqrt[3]{15 y z^{2}} {\sqrt[3]{15 y z^{2}}}^{2}}$

Raise $\sqrt[3]{15 y z^{2}}$ to the power of $1$ .

$\frac{\sqrt[3]{7 x^{2}} {\sqrt[3]{15 y z^{2}}}^{2}}{{\sqrt[3]{15 y z^{2}}}^{1} {\sqrt[3]{15 y z^{2}}}^{2}}$

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

$\frac{\sqrt[3]{7 x^{2}} {\sqrt[3]{15 y z^{2}}}^{2}}{{\sqrt[3]{15 y z^{2}}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{\sqrt[3]{7 x^{2}} {\sqrt[3]{15 y z^{2}}}^{2}}{{\sqrt[3]{15 y z^{2}}}^{3}}$

Rewrite ${\sqrt[3]{15 y z^{2}}}^{3}$ as $15 y z^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{15 y z^{2}}$ as ${(15 y z^{2})}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{7 x^{2}} {\sqrt[3]{15 y z^{2}}}^{2}}{{({(15 y z^{2})}^{\frac{1}{3}})}^{3}}$

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

$\frac{\sqrt[3]{7 x^{2}} {\sqrt[3]{15 y z^{2}}}^{2}}{{(15 y z^{2})}^{\frac{1}{3} \cdot 3}}$

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

$\frac{\sqrt[3]{7 x^{2}} {\sqrt[3]{15 y z^{2}}}^{2}}{{(15 y z^{2})}^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{\sqrt[3]{7 x^{2}} {\sqrt[3]{15 y z^{2}}}^{2}}{{(15 y z^{2})}^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{\sqrt[3]{7 x^{2}} {\sqrt[3]{15 y z^{2}}}^{2}}{{(15 y z^{2})}^{1}}$

Simplify.

$\frac{\sqrt[3]{7 x^{2}} {\sqrt[3]{15 y z^{2}}}^{2}}{15 y z^{2}}$

Step 4

Simplify the numerator.

Rewrite ${\sqrt[3]{15 y z^{2}}}^{2}$ as ${({(15 y z^{2})}^{2})}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{7 x^{2}} \sqrt[3]{{(15 y z^{2})}^{2}}}{15 y z^{2}}$

Apply the product rule to $15 y z^{2}$ .

$\frac{\sqrt[3]{7 x^{2}} \sqrt[3]{{(15 y)}^{2} {(z^{2})}^{2}}}{15 y z^{2}}$

Apply the product rule to $15 y$ .

$\frac{\sqrt[3]{7 x^{2}} \sqrt[3]{15^{2} y^{2} {(z^{2})}^{2}}}{15 y z^{2}}$

Raise $15$ to the power of $2$ .

$\frac{\sqrt[3]{7 x^{2}} \sqrt[3]{225 y^{2} {(z^{2})}^{2}}}{15 y z^{2}}$

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

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

$\frac{\sqrt[3]{7 x^{2}} \sqrt[3]{225 y^{2} z^{2 \cdot 2}}}{15 y z^{2}}$

Multiply $2$ by $2$ .

$\frac{\sqrt[3]{7 x^{2}} \sqrt[3]{225 y^{2} z^{4}}}{15 y z^{2}}$

Rewrite $225 y^{2} z^{4}$ as $z^{3} \cdot (225 y^{2} z)$ .

Factor out $z^{3}$ .

$\frac{\sqrt[3]{7 x^{2}} \sqrt[3]{225 y^{2} (z^{3} z)}}{15 y z^{2}}$

Move $y^{2}$ .

$\frac{\sqrt[3]{7 x^{2}} \sqrt[3]{225 z^{3} y^{2} z}}{15 y z^{2}}$

Reorder $225$ and $z^{3}$ .

$\frac{\sqrt[3]{7 x^{2}} \sqrt[3]{z^{3} \cdot 225 y^{2} z}}{15 y z^{2}}$

Add parentheses.

$\frac{\sqrt[3]{7 x^{2}} \sqrt[3]{z^{3} \cdot 225 (y^{2} z)}}{15 y z^{2}}$

Add parentheses.

$\frac{\sqrt[3]{7 x^{2}} \sqrt[3]{z^{3} \cdot (225 y^{2} z)}}{15 y z^{2}}$

Pull terms out from under the radical.

$\frac{\sqrt[3]{7 x^{2}} z \sqrt[3]{225 y^{2} z}}{15 y z^{2}}$

Combine exponents.

Combine using the product rule for radicals.

$\frac{z \sqrt[3]{7 x^{2} (225 y^{2} z)}}{15 y z^{2}}$

Multiply $225$ by $7$ .

$\frac{z \sqrt[3]{1575 x^{2} y^{2} z}}{15 y z^{2}}$

Step 5

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

Factor $z$ out of $z \sqrt[3]{1575 x^{2} y^{2} z}$ .

$\frac{z (\sqrt[3]{1575 x^{2} y^{2} z})}{15 y z^{2}}$

Cancel the common factors.

Factor $z$ out of $15 y z^{2}$ .

$\frac{z (\sqrt[3]{1575 x^{2} y^{2} z})}{z (15 y z)}$

Cancel the common factor.

$\frac{z \sqrt[3]{1575 x^{2} y^{2} z}}{z (15 y z)}$

Rewrite the expression.

$\frac{\sqrt[3]{1575 x^{2} y^{2} z}}{15 y z}$

Simplify cube root of (7x^2)/(15yz^2)

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