Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $x^{5} \sqrt[3]{x^{2}}$ as ${(x^{2})}^{2} (x \sqrt[3]{x^{2}})$ .

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Step 1.1.1

Factor out $x^{4}$ .

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

Step 1.1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 1.1.3

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

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

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Step 1.3.1

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

$\frac{\sqrt[5]{x^{3 + 2} \sqrt{x \sqrt[3]{x^{2}}}}}{\sqrt[3]{x^{2} \sqrt[5]{x^{3} \sqrt{x^{7}}}}}$

Step 1.3.2

Add $3$ and $2$ .

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

Step 1.4

Pull terms out from under the radical.

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

Step 1.5

Rewrite $\sqrt[5]{\sqrt{x \sqrt[3]{x^{2}}}}$ as $\sqrt[10]{x \sqrt[3]{x^{2}}}$ .

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[3]{x^{2} \sqrt[5]{x^{3} \sqrt{x^{7}}}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $x^{7}$ as ${(x^{3})}^{2} x$ .

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Step 2.1.1

Factor out $x^{6}$ .

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[3]{x^{2} \sqrt[5]{x^{3} \sqrt{x^{6} x}}}}$

Step 2.1.2

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[3]{x^{2} \sqrt[5]{x^{3} \sqrt{{(x^{3})}^{2} x}}}}$

Step 2.2

Pull terms out from under the radical.

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[3]{x^{2} \sqrt[5]{x^{3} x^{3} \sqrt{x}}}}$

Step 2.3

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

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Step 2.3.1

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

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[3]{x^{2} \sqrt[5]{x^{3 + 3} \sqrt{x}}}}$

Step 2.3.2

Add $3$ and $3$ .

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[3]{x^{2} \sqrt[5]{x^{6} \sqrt{x}}}}$

Step 2.4

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

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Step 2.4.1

Factor out $x^{5}$ .

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[3]{x^{2} \sqrt[5]{x^{5} x \sqrt{x}}}}$

Step 2.4.2

Add parentheses.

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[3]{x^{2} \sqrt[5]{x^{5} (x \sqrt{x})}}}$

Step 2.5

Pull terms out from under the radical.

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[3]{x^{2} x \sqrt[5]{x \sqrt{x}}}}$

Step 2.6

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.6.1

Multiply $x^{2}$ by $x$ .

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Step 2.6.1.1

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

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[3]{x^{2} x^{1} \sqrt[5]{x \sqrt{x}}}}$

Step 2.6.1.2

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

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[3]{x^{2 + 1} \sqrt[5]{x \sqrt{x}}}}$

Step 2.6.2

Add $2$ and $1$ .

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[3]{x^{3} \sqrt[5]{x \sqrt{x}}}}$

Step 2.7

Pull terms out from under the radical.

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{x \sqrt[3]{\sqrt[5]{x \sqrt{x}}}}$

Step 2.8

Rewrite $\sqrt[3]{\sqrt[5]{x \sqrt{x}}}$ as $\sqrt[15]{x \sqrt{x}}$ .

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{x \sqrt[15]{x \sqrt{x}}}$

Step 3

Cancel the common factor of $x$ .

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Step 3.1

Cancel the common factor.

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}}}}{x \sqrt[15]{x \sqrt{x}}}$

Step 3.2

Rewrite the expression.

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[15]{x \sqrt{x}}}$

Step 4

Multiply $\frac{\sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[15]{x \sqrt{x}}}$ by $\frac{{\sqrt[15]{x \sqrt{x}}}^{14}}{{\sqrt[15]{x \sqrt{x}}}^{14}}$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[15]{x \sqrt{x}}} \cdot \frac{{\sqrt[15]{x \sqrt{x}}}^{14}}{{\sqrt[15]{x \sqrt{x}}}^{14}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{\sqrt[10]{x \sqrt[3]{x^{2}}}}{\sqrt[15]{x \sqrt{x}}}$ by $\frac{{\sqrt[15]{x \sqrt{x}}}^{14}}{{\sqrt[15]{x \sqrt{x}}}^{14}}$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} {\sqrt[15]{x \sqrt{x}}}^{14}}{\sqrt[15]{x \sqrt{x}} {\sqrt[15]{x \sqrt{x}}}^{14}}$

Step 5.2

Raise $\sqrt[15]{x \sqrt{x}}$ to the power of $1$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} {\sqrt[15]{x \sqrt{x}}}^{14}}{{\sqrt[15]{x \sqrt{x}}}^{1} {\sqrt[15]{x \sqrt{x}}}^{14}}$

Step 5.3

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

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} {\sqrt[15]{x \sqrt{x}}}^{14}}{{\sqrt[15]{x \sqrt{x}}}^{1 + 14}}$

Step 5.4

Add $1$ and $14$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} {\sqrt[15]{x \sqrt{x}}}^{14}}{{\sqrt[15]{x \sqrt{x}}}^{15}}$

Step 5.5

Rewrite ${\sqrt[15]{x \sqrt{x}}}^{15}$ as $x \sqrt{x}$ .

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Step 5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[15]{x \sqrt{x}}$ as ${(x \sqrt{x})}^{\frac{1}{15}}$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} {\sqrt[15]{x \sqrt{x}}}^{14}}{{({(x \sqrt{x})}^{\frac{1}{15}})}^{15}}$

Step 5.5.2

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

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} {\sqrt[15]{x \sqrt{x}}}^{14}}{{(x \sqrt{x})}^{\frac{1}{15} \cdot 15}}$

Step 5.5.3

Combine $\frac{1}{15}$ and $15$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} {\sqrt[15]{x \sqrt{x}}}^{14}}{{(x \sqrt{x})}^{\frac{15}{15}}}$

Step 5.5.4

Cancel the common factor of $15$ .

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Step 5.5.4.1

Cancel the common factor.

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} {\sqrt[15]{x \sqrt{x}}}^{14}}{{(x \sqrt{x})}^{\frac{15}{15}}}$

Step 5.5.4.2

Rewrite the expression.

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} {\sqrt[15]{x \sqrt{x}}}^{14}}{{(x \sqrt{x})}^{1}}$

Step 5.5.5

Simplify.

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} {\sqrt[15]{x \sqrt{x}}}^{14}}{x \sqrt{x}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite ${\sqrt[15]{x \sqrt{x}}}^{14}$ as $\sqrt[15]{{(x \sqrt{x})}^{14}}$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} \sqrt[15]{{(x \sqrt{x})}^{14}}}{x \sqrt{x}}$

Step 6.2

Apply the product rule to $x \sqrt{x}$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} \sqrt[15]{x^{14} {\sqrt{x}}^{14}}}{x \sqrt{x}}$

Step 6.3

Rewrite ${\sqrt{x}}^{14}$ as $x^{7}$ .

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Step 6.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} \sqrt[15]{x^{14} {(x^{\frac{1}{2}})}^{14}}}{x \sqrt{x}}$

Step 6.3.2

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

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} \sqrt[15]{x^{14} x^{\frac{1}{2} \cdot 14}}}{x \sqrt{x}}$

Step 6.3.3

Combine $\frac{1}{2}$ and $14$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} \sqrt[15]{x^{14} x^{\frac{14}{2}}}}{x \sqrt{x}}$

Step 6.3.4

Cancel the common factor of $14$ and $2$ .

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Step 6.3.4.1

Factor $2$ out of $14$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} \sqrt[15]{x^{14} x^{\frac{2 \cdot 7}{2}}}}{x \sqrt{x}}$

Step 6.3.4.2

Cancel the common factors.

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Step 6.3.4.2.1

Factor $2$ out of $2$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} \sqrt[15]{x^{14} x^{\frac{2 \cdot 7}{2 (1)}}}}{x \sqrt{x}}$

Step 6.3.4.2.2

Cancel the common factor.

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} \sqrt[15]{x^{14} x^{\frac{2 \cdot 7}{2 \cdot 1}}}}{x \sqrt{x}}$

Step 6.3.4.2.3

Rewrite the expression.

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} \sqrt[15]{x^{14} x^{\frac{7}{1}}}}{x \sqrt{x}}$

Step 6.3.4.2.4

Divide $7$ by $1$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} \sqrt[15]{x^{14} x^{7}}}{x \sqrt{x}}$

Step 6.4

Multiply $x^{14}$ by $x^{7}$ by adding the exponents.

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Step 6.4.1

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

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} \sqrt[15]{x^{14 + 7}}}{x \sqrt{x}}$

Step 6.4.2

Add $14$ and $7$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} \sqrt[15]{x^{21}}}{x \sqrt{x}}$

Step 6.5

Factor out $x^{15}$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} \sqrt[15]{x^{15} x^{6}}}{x \sqrt{x}}$

Step 6.6

Pull terms out from under the radical.

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} (x \sqrt[15]{x^{6}})}{x \sqrt{x}}$

Step 6.7

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} (x \sqrt[15]{{(x^{2})}^{3}})}{x \sqrt{x}}$

Step 6.8

Rewrite $\sqrt[15]{{(x^{2})}^{3}}$ as $\sqrt[5]{\sqrt[3]{{(x^{2})}^{3}}}$ .

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} (x \sqrt[5]{\sqrt[3]{{(x^{2})}^{3}}})}{x \sqrt{x}}$

Step 6.9

Pull terms out from under the radical, assuming real numbers.

$\frac{\sqrt[10]{x \sqrt[3]{x^{2}}} x \sqrt[5]{x^{2}}}{x \sqrt{x}}$

Step 6.10

Combine exponents.

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Step 6.10.1

Rewrite the expression using the least common index of $10$ .

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Step 6.10.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{2}}$ as $x^{\frac{2}{5}}$ .

$\frac{x (\sqrt[10]{x \sqrt[3]{x^{2}}} x^{\frac{2}{5}})}{x \sqrt{x}}$

Step 6.10.1.2

Rewrite $x^{\frac{2}{5}}$ as $x^{\frac{4}{10}}$ .

$\frac{x (\sqrt[10]{x \sqrt[3]{x^{2}}} x^{\frac{4}{10}})}{x \sqrt{x}}$

Step 6.10.1.3

Rewrite $x^{\frac{4}{10}}$ as $\sqrt[10]{x^{4}}$ .

$\frac{x (\sqrt[10]{x \sqrt[3]{x^{2}}} \sqrt[10]{x^{4}})}{x \sqrt{x}}$

Step 6.10.2

Combine using the product rule for radicals.

$\frac{x \sqrt[10]{x \sqrt[3]{x^{2}} x^{4}}}{x \sqrt{x}}$

Step 6.10.3

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

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Step 6.10.3.1

Move $x^{4}$ .

$\frac{x \sqrt[10]{x^{4} x \sqrt[3]{x^{2}}}}{x \sqrt{x}}$

Step 6.10.3.2

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

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Step 6.10.3.2.1

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

$\frac{x \sqrt[10]{x^{4} x^{1} \sqrt[3]{x^{2}}}}{x \sqrt{x}}$

Step 6.10.3.2.2

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

$\frac{x \sqrt[10]{x^{4 + 1} \sqrt[3]{x^{2}}}}{x \sqrt{x}}$

Step 6.10.3.3

Add $4$ and $1$ .

$\frac{x \sqrt[10]{x^{5} \sqrt[3]{x^{2}}}}{x \sqrt{x}}$

Step 7

Cancel the common factor of $x$ .

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Step 7.1

Cancel the common factor.

$\frac{x \sqrt[10]{x^{5} \sqrt[3]{x^{2}}}}{x \sqrt{x}}$

Step 7.2

Rewrite the expression.

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}}}{\sqrt{x}}$

Step 8

Multiply $\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 9.2

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}$

Step 9.3

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}$

Step 9.4

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

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 9.5

Add $1$ and $1$ .

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}} \sqrt{x}}{{\sqrt{x}}^{2}}$

Step 9.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}$

Step 9.6.2

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

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

Step 9.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 9.6.4.2

Rewrite the expression.

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}} \sqrt{x}}{x^{1}}$

Step 9.6.5

Simplify.

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}} \sqrt{x}}{x}$

Step 10

Simplify the numerator.

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Step 10.1

Rewrite the expression using the least common index of $10$ .

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Step 10.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}} x^{\frac{1}{2}}}{x}$

Step 10.1.2

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{5}{10}}$ .

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}} x^{\frac{5}{10}}}{x}$

Step 10.1.3

Rewrite $x^{\frac{5}{10}}$ as $\sqrt[10]{x^{5}}$ .

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}}} \sqrt[10]{x^{5}}}{x}$

Step 10.2

Combine using the product rule for radicals.

$\frac{\sqrt[10]{x^{5} \sqrt[3]{x^{2}} x^{5}}}{x}$

Step 10.3

Multiply $x^{5}$ by $x^{5}$ by adding the exponents.

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Step 10.3.1

Move $x^{5}$ .

$\frac{\sqrt[10]{x^{5} x^{5} \sqrt[3]{x^{2}}}}{x}$

Step 10.3.2

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

$\frac{\sqrt[10]{x^{5 + 5} \sqrt[3]{x^{2}}}}{x}$

Step 10.3.3

Add $5$ and $5$ .

$\frac{\sqrt[10]{x^{10} \sqrt[3]{x^{2}}}}{x}$

Step 11

Simplify the numerator.

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Step 11.1

Pull terms out from under the radical.

$\frac{x \sqrt[10]{\sqrt[3]{x^{2}}}}{x}$

Step 11.2

Rewrite $\sqrt[10]{\sqrt[3]{x^{2}}}$ as $\sqrt[30]{x^{2}}$ .

$\frac{x \sqrt[30]{x^{2}}}{x}$

Step 11.3

Rewrite $\sqrt[30]{x^{2}}$ as $\sqrt[15]{\sqrt{x^{2}}}$ .

$\frac{x \sqrt[15]{\sqrt{x^{2}}}}{x}$

Step 11.4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{x \sqrt[15]{x}}{x}$

Step 12

Cancel the common factor of $x$ .

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Step 12.1

Cancel the common factor.

$\frac{x \sqrt[15]{x}}{x}$

Step 12.2

Divide $\sqrt[15]{x}$ by $1$ .

$\sqrt[15]{x}$