Algebra Examples

$\frac{9 x^{2}}{\sqrt[5]{27 x^{6}}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $27 x^{6}$ as $x^{5} \cdot (27 x)$ .

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Step 1.1.1

Factor out $x^{5}$ .

$\frac{9 x^{2}}{\sqrt[5]{27 (x^{5} x)}}$

Step 1.1.2

Reorder $27$ and $x^{5}$ .

$\frac{9 x^{2}}{\sqrt[5]{x^{5} \cdot 27 x}}$

Step 1.1.3

Add parentheses.

$\frac{9 x^{2}}{\sqrt[5]{x^{5} \cdot (27 x)}}$

Step 1.2

Pull terms out from under the radical.

$\frac{9 x^{2}}{x \sqrt[5]{27 x}}$

Step 2

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

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Step 2.1

Factor $x$ out of $9 x^{2}$ .

$\frac{x (9 x)}{x \sqrt[5]{27 x}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $x$ out of $x \sqrt[5]{27 x}$ .

$\frac{x (9 x)}{x (\sqrt[5]{27 x})}$

Step 2.2.2

Cancel the common factor.

$\frac{x (9 x)}{x \sqrt[5]{27 x}}$

Step 2.2.3

Rewrite the expression.

$\frac{9 x}{\sqrt[5]{27 x}}$

Step 3

Multiply $\frac{9 x}{\sqrt[5]{27 x}}$ by $\frac{{\sqrt[5]{27 x}}^{4}}{{\sqrt[5]{27 x}}^{4}}$ .

$\frac{9 x}{\sqrt[5]{27 x}} \cdot \frac{{\sqrt[5]{27 x}}^{4}}{{\sqrt[5]{27 x}}^{4}}$

Step 4

Simplify terms.

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Step 4.1

Combine and simplify the denominator.

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Step 4.1.1

Multiply $\frac{9 x}{\sqrt[5]{27 x}}$ by $\frac{{\sqrt[5]{27 x}}^{4}}{{\sqrt[5]{27 x}}^{4}}$ .

$\frac{9 x {\sqrt[5]{27 x}}^{4}}{\sqrt[5]{27 x} {\sqrt[5]{27 x}}^{4}}$

Step 4.1.2

Raise $\sqrt[5]{27 x}$ to the power of $1$ .

$\frac{9 x {\sqrt[5]{27 x}}^{4}}{{\sqrt[5]{27 x}}^{1} {\sqrt[5]{27 x}}^{4}}$

Step 4.1.3

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

$\frac{9 x {\sqrt[5]{27 x}}^{4}}{{\sqrt[5]{27 x}}^{1 + 4}}$

Step 4.1.4

Add $1$ and $4$ .

$\frac{9 x {\sqrt[5]{27 x}}^{4}}{{\sqrt[5]{27 x}}^{5}}$

Step 4.1.5

Rewrite ${\sqrt[5]{27 x}}^{5}$ as $27 x$ .

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Step 4.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{27 x}$ as ${(27 x)}^{\frac{1}{5}}$ .

$\frac{9 x {\sqrt[5]{27 x}}^{4}}{{({(27 x)}^{\frac{1}{5}})}^{5}}$

Step 4.1.5.2

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

$\frac{9 x {\sqrt[5]{27 x}}^{4}}{{(27 x)}^{\frac{1}{5} \cdot 5}}$

Step 4.1.5.3

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

$\frac{9 x {\sqrt[5]{27 x}}^{4}}{{(27 x)}^{\frac{5}{5}}}$

Step 4.1.5.4

Cancel the common factor of $5$ .

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Step 4.1.5.4.1

Cancel the common factor.

$\frac{9 x {\sqrt[5]{27 x}}^{4}}{{(27 x)}^{\frac{5}{5}}}$

Step 4.1.5.4.2

Rewrite the expression.

$\frac{9 x {\sqrt[5]{27 x}}^{4}}{{(27 x)}^{1}}$

Step 4.1.5.5

Simplify.

$\frac{9 x {\sqrt[5]{27 x}}^{4}}{27 x}$

Step 4.2

Cancel the common factor of $9$ and $27$ .

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Step 4.2.1

Factor $9$ out of $9 x {\sqrt[5]{27 x}}^{4}$ .

$\frac{9 (x {\sqrt[5]{27 x}}^{4})}{27 x}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $9$ out of $27 x$ .

$\frac{9 (x {\sqrt[5]{27 x}}^{4})}{9 (3 x)}$

Step 4.2.2.2

Cancel the common factor.

$\frac{9 (x {\sqrt[5]{27 x}}^{4})}{9 (3 x)}$

Step 4.2.2.3

Rewrite the expression.

$\frac{x {\sqrt[5]{27 x}}^{4}}{3 x}$

Step 4.3

Cancel the common factor of $x$ .

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Step 4.3.1

Cancel the common factor.

$\frac{x {\sqrt[5]{27 x}}^{4}}{3 x}$

Step 4.3.2

Rewrite the expression.

$\frac{{\sqrt[5]{27 x}}^{4}}{3}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt[5]{27 x}}^{4}$ as $\sqrt[5]{{(27 x)}^{4}}$ .

$\frac{\sqrt[5]{{(27 x)}^{4}}}{3}$

Step 5.2

Apply the product rule to $27 x$ .

$\frac{\sqrt[5]{27^{4} x^{4}}}{3}$

Step 5.3

Raise $27$ to the power of $4$ .

$\frac{\sqrt[5]{531441 x^{4}}}{3}$

Step 5.4

Rewrite $531441 x^{4}$ as $9^{5} \cdot (9 x^{4})$ .

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Step 5.4.1

Factor $59049$ out of $531441$ .

$\frac{\sqrt[5]{59049 (9) x^{4}}}{3}$

Step 5.4.2

Rewrite $59049$ as $9^{5}$ .

$\frac{\sqrt[5]{9^{5} \cdot 9 x^{4}}}{3}$

Step 5.4.3

Add parentheses.

$\frac{\sqrt[5]{9^{5} \cdot (9 x^{4})}}{3}$

Step 5.5

Pull terms out from under the radical.

$\frac{9 \sqrt[5]{9 x^{4}}}{3}$

Step 6

Cancel the common factor of $9$ and $3$ .

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Step 6.1

Factor $3$ out of $9 \sqrt[5]{9 x^{4}}$ .

$\frac{3 (3 \sqrt[5]{9 x^{4}})}{3}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $3$ out of $3$ .

$\frac{3 (3 \sqrt[5]{9 x^{4}})}{3 (1)}$

Step 6.2.2

Cancel the common factor.

$\frac{3 (3 \sqrt[5]{9 x^{4}})}{3 \cdot 1}$

Step 6.2.3

Rewrite the expression.

$\frac{3 \sqrt[5]{9 x^{4}}}{1}$

Step 6.2.4

Divide $3 \sqrt[5]{9 x^{4}}$ by $1$ .

$3 \sqrt[5]{9 x^{4}}$