Algebra Examples

$\sqrt[5]{\frac{243 x^{7}}{y^{14} z^{21}}}$

Step 1

Rewrite $\frac{243 x^{7}}{y^{14} z^{21}}$ as ${(\frac{3 x}{y^{2} z^{4}})}^{5} \frac{x^{2}}{y^{4} z}$ .

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Step 1.1

Factor the perfect power ${(3 x)}^{5}$ out of $243 x^{7}$ .

$\sqrt[5]{\frac{{(3 x)}^{5} x^{2}}{y^{14} z^{21}}}$

Step 1.2

Factor the perfect power ${(y^{2} z^{4})}^{5}$ out of $y^{14} z^{21}$ .

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

Step 1.3

Rearrange the fraction $\frac{{(3 x)}^{5} x^{2}}{{(y^{2} z^{4})}^{5} (y^{4} z)}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite $\sqrt[5]{\frac{x^{2}}{y^{4} z}}$ as $\frac{\sqrt[5]{x^{2}}}{\sqrt[5]{y^{4} z}}$ .

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

Step 4

Combine.

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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Step 6.1

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

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

Step 6.2

Move $\sqrt[5]{y^{4} z}$ .

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

Step 6.3

Raise $\sqrt[5]{y^{4} z}$ to the power of $1$ .

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

Step 6.4

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

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

Step 6.5

Add $1$ and $4$ .

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

Step 6.6

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

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y^{4} z}$ as ${(y^{4} z)}^{\frac{1}{5}}$ .

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

Step 6.6.2

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

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

Step 6.6.3

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

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

Step 6.6.4

Cancel the common factor of $5$ .

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Step 6.6.4.1

Cancel the common factor.

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

Step 6.6.4.2

Rewrite the expression.

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

Step 6.6.5

Simplify.

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

Step 7

Multiply $y^{2}$ by $y^{4}$ by adding the exponents.

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Step 7.1

Move $y^{4}$ .

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

Step 7.2

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

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

Step 7.3

Add $4$ and $2$ .

$\frac{3 x \sqrt[5]{x^{2}} {\sqrt[5]{y^{4} z}}^{4}}{y^{6} z^{4} z}$

Step 8

Multiply $z^{4}$ by $z$ by adding the exponents.

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Step 8.1

Move $z$ .

$\frac{3 x \sqrt[5]{x^{2}} {\sqrt[5]{y^{4} z}}^{4}}{y^{6} (z \cdot z^{4})}$

Step 8.2

Multiply $z$ by $z^{4}$ .

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Step 8.2.1

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

$\frac{3 x \sqrt[5]{x^{2}} {\sqrt[5]{y^{4} z}}^{4}}{y^{6} (z^{1} z^{4})}$

Step 8.2.2

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

$\frac{3 x \sqrt[5]{x^{2}} {\sqrt[5]{y^{4} z}}^{4}}{y^{6} z^{1 + 4}}$

Step 8.3

Add $1$ and $4$ .

$\frac{3 x \sqrt[5]{x^{2}} {\sqrt[5]{y^{4} z}}^{4}}{y^{6} z^{5}}$

Step 9

Simplify the numerator.

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Step 9.1

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

$\frac{3 x \sqrt[5]{x^{2}} \sqrt[5]{{(y^{4} z)}^{4}}}{y^{6} z^{5}}$

Step 9.2

Apply the product rule to $y^{4} z$ .

$\frac{3 x \sqrt[5]{x^{2}} \sqrt[5]{{(y^{4})}^{4} z^{4}}}{y^{6} z^{5}}$

Step 9.3

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

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Step 9.3.1

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

$\frac{3 x \sqrt[5]{x^{2}} \sqrt[5]{y^{4 \cdot 4} z^{4}}}{y^{6} z^{5}}$

Step 9.3.2

Multiply $4$ by $4$ .

$\frac{3 x \sqrt[5]{x^{2}} \sqrt[5]{y^{16} z^{4}}}{y^{6} z^{5}}$

Step 9.4

Rewrite $y^{16} z^{4}$ as ${(y^{3})}^{5} (y z^{4})$ .

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Step 9.4.1

Factor out $y^{15}$ .

$\frac{3 x \sqrt[5]{x^{2}} \sqrt[5]{y^{15} y z^{4}}}{y^{6} z^{5}}$

Step 9.4.2

Rewrite $y^{15}$ as ${(y^{3})}^{5}$ .

$\frac{3 x \sqrt[5]{x^{2}} \sqrt[5]{{(y^{3})}^{5} y z^{4}}}{y^{6} z^{5}}$

Step 9.4.3

Add parentheses.

$\frac{3 x \sqrt[5]{x^{2}} \sqrt[5]{{(y^{3})}^{5} (y z^{4})}}{y^{6} z^{5}}$

Step 9.5

Pull terms out from under the radical.

$\frac{3 x \sqrt[5]{x^{2}} y^{3} \sqrt[5]{y z^{4}}}{y^{6} z^{5}}$

Step 9.6

Combine using the product rule for radicals.

$\frac{3 x y^{3} \sqrt[5]{x^{2} y z^{4}}}{y^{6} z^{5}}$

Step 10

Cancel the common factor of $y^{3}$ and $y^{6}$ .

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Step 10.1

Factor $y^{3}$ out of $3 x y^{3} \sqrt[5]{x^{2} y z^{4}}$ .

$\frac{y^{3} (3 x \sqrt[5]{x^{2} y z^{4}})}{y^{6} z^{5}}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $y^{3}$ out of $y^{6} z^{5}$ .

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

Step 10.2.2

Cancel the common factor.

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

Step 10.2.3

Rewrite the expression.

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