Simplify the Radical Expression ( fourth root of z^5)/( fifth root of z)

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

Simplify the numerator.

Factor out $z^{4}$ .

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

Pull terms out from under the radical.

$\frac{| z | \sqrt[4]{z}}{\sqrt[5]{z}}$

Multiply $\frac{| z | \sqrt[4]{z}}{\sqrt[5]{z}}$ by $\frac{{\sqrt[5]{z}}^{4}}{{\sqrt[5]{z}}^{4}}$ .

$\frac{| z | \sqrt[4]{z}}{\sqrt[5]{z}} \cdot \frac{{\sqrt[5]{z}}^{4}}{{\sqrt[5]{z}}^{4}}$

Combine and simplify the denominator.

Multiply $\frac{| z | \sqrt[4]{z}}{\sqrt[5]{z}}$ by $\frac{{\sqrt[5]{z}}^{4}}{{\sqrt[5]{z}}^{4}}$ .

$\frac{| z | \sqrt[4]{z} {\sqrt[5]{z}}^{4}}{\sqrt[5]{z} {\sqrt[5]{z}}^{4}}$

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

$\frac{| z | \sqrt[4]{z} {\sqrt[5]{z}}^{4}}{{\sqrt[5]{z}}^{1} {\sqrt[5]{z}}^{4}}$

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

$\frac{| z | \sqrt[4]{z} {\sqrt[5]{z}}^{4}}{{\sqrt[5]{z}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{| z | \sqrt[4]{z} {\sqrt[5]{z}}^{4}}{{\sqrt[5]{z}}^{5}}$

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

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

$\frac{| z | \sqrt[4]{z} {\sqrt[5]{z}}^{4}}{{(z^{\frac{1}{5}})}^{5}}$

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

$\frac{| z | \sqrt[4]{z} {\sqrt[5]{z}}^{4}}{z^{\frac{1}{5} \cdot 5}}$

Combine $\frac{1}{5}$ and $5$ .

$\frac{| z | \sqrt[4]{z} {\sqrt[5]{z}}^{4}}{z^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{| z | \sqrt[4]{z} {\sqrt[5]{z}}^{4}}{z^{\frac{5}{5}}}$

Rewrite the expression.

$\frac{| z | \sqrt[4]{z} {\sqrt[5]{z}}^{4}}{z^{1}}$

Simplify.

$\frac{| z | \sqrt[4]{z} {\sqrt[5]{z}}^{4}}{z}$

Simplify the numerator.

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

$\frac{| z | \sqrt[4]{z} \sqrt[5]{z^{4}}}{z}$

Combine exponents.

Rewrite the expression using the least common index of $20$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{z^{4}}$ as $z^{\frac{4}{5}}$ .

$\frac{| z | (z^{\frac{4}{5}} \sqrt[4]{z})}{z}$

Rewrite $z^{\frac{4}{5}}$ as $z^{\frac{16}{20}}$ .

$\frac{| z | (z^{\frac{16}{20}} \sqrt[4]{z})}{z}$

Rewrite $z^{\frac{16}{20}}$ as $\sqrt[20]{z^{16}}$ .

$\frac{| z | (\sqrt[20]{z^{16}} \sqrt[4]{z})}{z}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{z}$ as $z^{\frac{1}{4}}$ .

$\frac{| z | (\sqrt[20]{z^{16}} z^{\frac{1}{4}})}{z}$

Rewrite $z^{\frac{1}{4}}$ as $z^{\frac{5}{20}}$ .

$\frac{| z | (\sqrt[20]{z^{16}} z^{\frac{5}{20}})}{z}$

Rewrite $z^{\frac{5}{20}}$ as $\sqrt[20]{z^{5}}$ .

$\frac{| z | (\sqrt[20]{z^{16}} \sqrt[20]{z^{5}})}{z}$

Combine using the product rule for radicals.

$\frac{| z | \sqrt[20]{z^{16} z^{5}}}{z}$

Multiply $z^{16}$ by $z^{5}$ by adding the exponents.

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

$\frac{| z | \sqrt[20]{z^{16 + 5}}}{z}$

Add $16$ and $5$ .

$\frac{| z | \sqrt[20]{z^{21}}}{z}$

Factor out $z^{20}$ .

$\frac{| z | \sqrt[20]{z^{20} z}}{z}$

Pull terms out from under the radical.

$\frac{| z | (| z | \sqrt[20]{z})}{z}$

To multiply absolute values, multiply the terms inside each absolute value.

$\frac{| z \cdot z | \sqrt[20]{z}}{z}$

Simplify the numerator.

Multiply $z$ by $z$ .

$\frac{| z^{2} | \sqrt[20]{z}}{z}$

Remove non-negative terms from the absolute value.

$\frac{z^{2} \sqrt[20]{z}}{z}$

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

Factor $z$ out of $z^{2} \sqrt[20]{z}$ .

$\frac{z (z \sqrt[20]{z})}{z}$

Cancel the common factors.

Raise $z$ to the power of $1$ .

$\frac{z (z \sqrt[20]{z})}{z^{1}}$

Factor $z$ out of $z^{1}$ .

$\frac{z (z \sqrt[20]{z})}{z \cdot 1}$

Cancel the common factor.

$\frac{z (z \sqrt[20]{z})}{z \cdot 1}$

Rewrite the expression.

$\frac{z \sqrt[20]{z}}{1}$

Divide $z \sqrt[20]{z}$ by $1$ .

$z \sqrt[20]{z}$

Simplify the Radical Expression ( fourth root of z^5)/( fifth root of z)

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