Simplify (w^7)/( fifth root of z^13)

$\frac{w^{7}}{\sqrt[5]{z^{13}}}$

Simplify the denominator.

Rewrite $z^{13}$ as ${(z^{2})}^{5} z^{3}$ .

Factor out $z^{10}$ .

$\frac{w^{7}}{\sqrt[5]{z^{10} z^{3}}}$

Rewrite $z^{10}$ as ${(z^{2})}^{5}$ .

$\frac{w^{7}}{\sqrt[5]{{(z^{2})}^{5} z^{3}}}$

Pull terms out from under the radical.

$\frac{w^{7}}{z^{2} \sqrt[5]{z^{3}}}$

Multiply $\frac{w^{7}}{z^{2} \sqrt[5]{z^{3}}}$ by $\frac{{\sqrt[5]{z^{3}}}^{4}}{{\sqrt[5]{z^{3}}}^{4}}$ .

$\frac{w^{7}}{z^{2} \sqrt[5]{z^{3}}} \cdot \frac{{\sqrt[5]{z^{3}}}^{4}}{{\sqrt[5]{z^{3}}}^{4}}$

Combine and simplify the denominator.

Multiply $\frac{w^{7}}{z^{2} \sqrt[5]{z^{3}}}$ by $\frac{{\sqrt[5]{z^{3}}}^{4}}{{\sqrt[5]{z^{3}}}^{4}}$ .

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2} \sqrt[5]{z^{3}} {\sqrt[5]{z^{3}}}^{4}}$

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

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2} (\sqrt[5]{z^{3}} {\sqrt[5]{z^{3}}}^{4})}$

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

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2} ({\sqrt[5]{z^{3}}}^{1} {\sqrt[5]{z^{3}}}^{4})}$

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

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2} {\sqrt[5]{z^{3}}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2} {\sqrt[5]{z^{3}}}^{5}}$

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

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

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2} {(z^{\frac{3}{5}})}^{5}}$

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

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2} z^{\frac{3}{5} \cdot 5}}$

Combine $\frac{3}{5}$ and $5$ .

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2} z^{\frac{3 \cdot 5}{5}}}$

Multiply $3$ by $5$ .

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2} z^{\frac{15}{5}}}$

Cancel the common factor of $15$ and $5$ .

Factor $5$ out of $15$ .

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2} z^{\frac{5 \cdot 3}{5}}}$

Cancel the common factors.

Factor $5$ out of $5$ .

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2} z^{\frac{5 \cdot 3}{5 (1)}}}$

Cancel the common factor.

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2} z^{\frac{5 \cdot 3}{5 \cdot 1}}}$

Rewrite the expression.

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2} z^{\frac{3}{1}}}$

Divide $3$ by $1$ .

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2} z^{3}}$

Multiply $z^{2}$ by $z^{3}$ by adding the exponents.

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

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{2 + 3}}$

Add $2$ and $3$ .

$\frac{w^{7} {\sqrt[5]{z^{3}}}^{4}}{z^{5}}$

Simplify the numerator.

Rewrite ${\sqrt[5]{z^{3}}}^{4}$ as ${({(z^{3})}^{4})}^{\frac{1}{5}}$ .

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

Multiply the exponents in ${(z^{3})}^{4}$ .

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

$\frac{w^{7} \sqrt[5]{z^{3 \cdot 4}}}{z^{5}}$

Multiply $3$ by $4$ .

$\frac{w^{7} \sqrt[5]{z^{12}}}{z^{5}}$

Rewrite $z^{12}$ as ${(z^{2})}^{5} z^{2}$ .

Factor out $z^{10}$ .

$\frac{w^{7} \sqrt[5]{z^{10} z^{2}}}{z^{5}}$

Rewrite $z^{10}$ as ${(z^{2})}^{5}$ .

$\frac{w^{7} \sqrt[5]{{(z^{2})}^{5} z^{2}}}{z^{5}}$

Pull terms out from under the radical.

$\frac{w^{7} z^{2} \sqrt[5]{z^{2}}}{z^{5}}$

Cancel the common factor of $z^{2}$ and $z^{5}$ .

Factor $z^{2}$ out of $w^{7} z^{2} \sqrt[5]{z^{2}}$ .

$\frac{z^{2} (w^{7} \sqrt[5]{z^{2}})}{z^{5}}$

Cancel the common factors.

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

$\frac{z^{2} (w^{7} \sqrt[5]{z^{2}})}{z^{2} z^{3}}$

Cancel the common factor.

$\frac{z^{2} (w^{7} \sqrt[5]{z^{2}})}{z^{2} z^{3}}$

Rewrite the expression.

$\frac{w^{7} \sqrt[5]{z^{2}}}{z^{3}}$

Simplify (w^7)/( fifth root of z^13)

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