Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Factor out $z^{4}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $4$ .

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Cancel the common factor of $5$ .

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Step 3.5.4.1

Cancel the common factor.

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

Step 3.5.4.2

Rewrite the expression.

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

Step 3.5.5

Simplify.

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[5]{z}}^{4}$ as $\sqrt[5]{z^{4}}$ .

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

Step 4.2

Combine exponents.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

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

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

Step 4.2.1.5

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

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

Step 4.2.1.6

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

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

Step 4.2.2

Combine using the product rule for radicals.

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

Step 4.2.3

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

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Step 4.2.3.1

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

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

Step 4.2.3.2

Add $16$ and $5$ .

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

Step 4.3

Factor out $z^{20}$ .

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

Step 4.4

Pull terms out from under the radical.

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

Step 5

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

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $z$ by $z$ .

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

Step 6.2

Remove non-negative terms from the absolute value.

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.2.3

Cancel the common factor.

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

Step 7.2.4

Rewrite the expression.

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

Step 7.2.5

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

$z \sqrt[20]{z}$