Simplify ( cube root of 5z^7b^4)/( cube root of 25xy)

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

Combine $\sqrt[3]{5 z^{7} b^{4}}$ and $\sqrt[3]{25 x y}$ into a single radical.

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

Reduce the expression $\frac{5 z^{7} b^{4}}{25 x y}$ by cancelling the common factors.

Factor $5$ out of $5 z^{7} b^{4}$ .

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

Factor $5$ out of $25 x y$ .

$\sqrt[3]{\frac{5 (z^{7} b^{4})}{5 (5 x y)}}$

Cancel the common factor.

$\sqrt[3]{\frac{5 (z^{7} b^{4})}{5 (5 x y)}}$

Rewrite the expression.

$\sqrt[3]{\frac{z^{7} b^{4}}{5 x y}}$

Rewrite $\sqrt[3]{\frac{z^{7} b^{4}}{5 x y}}$ as $\frac{\sqrt[3]{z^{7} b^{4}}}{\sqrt[3]{5 x y}}$ .

$\frac{\sqrt[3]{z^{7} b^{4}}}{\sqrt[3]{5 x y}}$

Simplify the numerator.

Rewrite $z^{7} b^{4}$ as ${(z^{2} b)}^{3} (z b)$ .

Factor out $z^{6}$ .

$\frac{\sqrt[3]{z^{6} z b^{4}}}{\sqrt[3]{5 x y}}$

Rewrite $z^{6}$ as ${(z^{2})}^{3}$ .

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

Factor out $b^{3}$ .

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

Move $z$ .

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

Rewrite ${(z^{2})}^{3} b^{3}$ as ${(z^{2} b)}^{3}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the numerator.

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

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

Apply the product rule to $5 x y$ .

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

Apply the product rule to $5 x$ .

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

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

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

Combine using the product rule for radicals.

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

Simplify ( cube root of 5z^7b^4)/( cube root of 25xy)

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