Simplify ( cube root of 36xy^2)/( cube root of 10xz)

$\frac{\sqrt[3]{36 x y^{2}}}{\sqrt[3]{10 x z}}$

Step 1

Combine $\sqrt[3]{36 x y^{2}}$ and $\sqrt[3]{10 x z}$ into a single radical.

$\sqrt[3]{\frac{36 x y^{2}}{10 x z}}$

Step 2

Reduce the expression $\frac{36 x y^{2}}{10 x z}$ by cancelling the common factors.

Factor $2$ out of $36 x y^{2}$ .

$\sqrt[3]{\frac{2 (18 x y^{2})}{10 x z}}$

Factor $2$ out of $10 x z$ .

$\sqrt[3]{\frac{2 (18 x y^{2})}{2 (5 x z)}}$

Cancel the common factor.

$\sqrt[3]{\frac{2 (18 x y^{2})}{2 (5 x z)}}$

Rewrite the expression.

$\sqrt[3]{\frac{18 x y^{2}}{5 x z}}$

Step 3

Cancel the common factor of $x$ .

Cancel the common factor.

$\sqrt[3]{\frac{18 x y^{2}}{5 x z}}$

Rewrite the expression.

$\sqrt[3]{\frac{18 y^{2}}{5 z}}$

Step 4

Rewrite $\sqrt[3]{\frac{18 y^{2}}{5 z}}$ as $\frac{\sqrt[3]{18 y^{2}}}{\sqrt[3]{5 z}}$ .

$\frac{\sqrt[3]{18 y^{2}}}{\sqrt[3]{5 z}}$

Step 5

Multiply $\frac{\sqrt[3]{18 y^{2}}}{\sqrt[3]{5 z}}$ by $\frac{{\sqrt[3]{5 z}}^{2}}{{\sqrt[3]{5 z}}^{2}}$ .

$\frac{\sqrt[3]{18 y^{2}}}{\sqrt[3]{5 z}} \cdot \frac{{\sqrt[3]{5 z}}^{2}}{{\sqrt[3]{5 z}}^{2}}$

Step 6

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{18 y^{2}}}{\sqrt[3]{5 z}}$ by $\frac{{\sqrt[3]{5 z}}^{2}}{{\sqrt[3]{5 z}}^{2}}$ .

$\frac{\sqrt[3]{18 y^{2}} {\sqrt[3]{5 z}}^{2}}{\sqrt[3]{5 z} {\sqrt[3]{5 z}}^{2}}$

Raise $\sqrt[3]{5 z}$ to the power of $1$ .

$\frac{\sqrt[3]{18 y^{2}} {\sqrt[3]{5 z}}^{2}}{{\sqrt[3]{5 z}}^{1} {\sqrt[3]{5 z}}^{2}}$

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

$\frac{\sqrt[3]{18 y^{2}} {\sqrt[3]{5 z}}^{2}}{{\sqrt[3]{5 z}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{\sqrt[3]{18 y^{2}} {\sqrt[3]{5 z}}^{2}}{{\sqrt[3]{5 z}}^{3}}$

Rewrite ${\sqrt[3]{5 z}}^{3}$ as $5 z$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 z}$ as ${(5 z)}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{18 y^{2}} {\sqrt[3]{5 z}}^{2}}{{({(5 z)}^{\frac{1}{3}})}^{3}}$

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

$\frac{\sqrt[3]{18 y^{2}} {\sqrt[3]{5 z}}^{2}}{{(5 z)}^{\frac{1}{3} \cdot 3}}$

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

$\frac{\sqrt[3]{18 y^{2}} {\sqrt[3]{5 z}}^{2}}{{(5 z)}^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{\sqrt[3]{18 y^{2}} {\sqrt[3]{5 z}}^{2}}{{(5 z)}^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{\sqrt[3]{18 y^{2}} {\sqrt[3]{5 z}}^{2}}{{(5 z)}^{1}}$

Simplify.

$\frac{\sqrt[3]{18 y^{2}} {\sqrt[3]{5 z}}^{2}}{5 z}$

Step 7

Simplify the numerator.

Rewrite ${\sqrt[3]{5 z}}^{2}$ as $\sqrt[3]{{(5 z)}^{2}}$ .

$\frac{\sqrt[3]{18 y^{2}} \sqrt[3]{{(5 z)}^{2}}}{5 z}$

Apply the product rule to $5 z$ .

$\frac{\sqrt[3]{18 y^{2}} \sqrt[3]{5^{2} z^{2}}}{5 z}$

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

$\frac{\sqrt[3]{18 y^{2}} \sqrt[3]{25 z^{2}}}{5 z}$

Step 8

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{\sqrt[3]{18 y^{2} (25 z^{2})}}{5 z}$

Multiply $25$ by $18$ .

$\frac{\sqrt[3]{450 y^{2} z^{2}}}{5 z}$

Simplify ( cube root of 36xy^2)/( cube root of 10xz)

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