Algebra Examples

$\frac{\sqrt[4]{x} \cdot \sqrt[4]{81 x}}{\sqrt[4]{16 x^{36}}}$

Step 1

Combine $\sqrt[4]{x}$ and $\sqrt[4]{16 x^{36}}$ into a single radical.

$\sqrt[4]{\frac{x}{16 x^{36}}} \cdot \sqrt[4]{81 x}$

Step 2

Reduce the expression $\frac{x}{16 x^{36}}$ by cancelling the common factors.

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Step 2.1

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

$\sqrt[4]{\frac{x^{1}}{16 x^{36}}} \cdot \sqrt[4]{81 x}$

Step 2.2

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

$\sqrt[4]{\frac{x \cdot 1}{16 x^{36}}} \cdot \sqrt[4]{81 x}$

Step 2.3

Factor $x$ out of $16 x^{36}$ .

$\sqrt[4]{\frac{x \cdot 1}{x (16 x^{35})}} \cdot \sqrt[4]{81 x}$

Step 2.4

Cancel the common factor.

$\sqrt[4]{\frac{x \cdot 1}{x (16 x^{35})}} \cdot \sqrt[4]{81 x}$

Step 2.5

Rewrite the expression.

$\sqrt[4]{\frac{1}{16 x^{35}}} \cdot \sqrt[4]{81 x}$

Step 3

Rewrite $\frac{1}{16 x^{35}}$ as ${(\frac{1}{2 x^{8}})}^{4} \frac{1}{x^{3}}$ .

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Step 3.1

Factor the perfect power $1^{4}$ out of $1$ .

$\sqrt[4]{\frac{1^{4} \cdot 1}{16 x^{35}}} \cdot \sqrt[4]{81 x}$

Step 3.2

Factor the perfect power ${(2 x^{8})}^{4}$ out of $16 x^{35}$ .

$\sqrt[4]{\frac{1^{4} \cdot 1}{{(2 x^{8})}^{4} x^{3}}} \cdot \sqrt[4]{81 x}$

Step 3.3

Rearrange the fraction $\frac{1^{4} \cdot 1}{{(2 x^{8})}^{4} x^{3}}$ .

$\sqrt[4]{{(\frac{1}{2 x^{8}})}^{4} \frac{1}{x^{3}}} \cdot \sqrt[4]{81 x}$

Step 4

Pull terms out from under the radical.

$\frac{1}{2 x^{8}} \sqrt[4]{\frac{1}{x^{3}}} \cdot \sqrt[4]{81 x}$

Step 5

Rewrite $\sqrt[4]{\frac{1}{x^{3}}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{x^{3}}}$ .

$\frac{1}{2 x^{8}} \cdot \frac{\sqrt[4]{1}}{\sqrt[4]{x^{3}}} \cdot \sqrt[4]{81 x}$

Step 6

Combine.

$\frac{1 \sqrt[4]{1}}{2 x^{8} \sqrt[4]{x^{3}}} \cdot \sqrt[4]{81 x}$

Step 7

Multiply $\sqrt[4]{1}$ by $1$ .

$\frac{\sqrt[4]{1}}{2 x^{8} \sqrt[4]{x^{3}}} \cdot \sqrt[4]{81 x}$

Step 8

Any root of $1$ is $1$ .

$\frac{1}{2 x^{8} \sqrt[4]{x^{3}}} \cdot \sqrt[4]{81 x}$

Step 9

Multiply $\frac{1}{2 x^{8} \sqrt[4]{x^{3}}}$ by $\frac{{\sqrt[4]{x^{3}}}^{3}}{{\sqrt[4]{x^{3}}}^{3}}$ .

$\frac{1}{2 x^{8} \sqrt[4]{x^{3}}} \cdot \frac{{\sqrt[4]{x^{3}}}^{3}}{{\sqrt[4]{x^{3}}}^{3}} \cdot \sqrt[4]{81 x}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{1}{2 x^{8} \sqrt[4]{x^{3}}}$ by $\frac{{\sqrt[4]{x^{3}}}^{3}}{{\sqrt[4]{x^{3}}}^{3}}$ .

$\frac{{\sqrt[4]{x^{3}}}^{3}}{2 x^{8} \sqrt[4]{x^{3}} {\sqrt[4]{x^{3}}}^{3}} \cdot \sqrt[4]{81 x}$

Step 10.2

Move $\sqrt[4]{x^{3}}$ .

$\frac{{\sqrt[4]{x^{3}}}^{3}}{2 x^{8} (\sqrt[4]{x^{3}} {\sqrt[4]{x^{3}}}^{3})} \cdot \sqrt[4]{81 x}$

Step 10.3

Raise $\sqrt[4]{x^{3}}$ to the power of $1$ .

$\frac{{\sqrt[4]{x^{3}}}^{3}}{2 x^{8} ({\sqrt[4]{x^{3}}}^{1} {\sqrt[4]{x^{3}}}^{3})} \cdot \sqrt[4]{81 x}$

Step 10.4

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

$\frac{{\sqrt[4]{x^{3}}}^{3}}{2 x^{8} {\sqrt[4]{x^{3}}}^{1 + 3}} \cdot \sqrt[4]{81 x}$

Step 10.5

Add $1$ and $3$ .

$\frac{{\sqrt[4]{x^{3}}}^{3}}{2 x^{8} {\sqrt[4]{x^{3}}}^{4}} \cdot \sqrt[4]{81 x}$

Step 10.6

Rewrite ${\sqrt[4]{x^{3}}}^{4}$ as $x^{3}$ .

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Step 10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

$\frac{{\sqrt[4]{x^{3}}}^{3}}{2 x^{8} {(x^{\frac{3}{4}})}^{4}} \cdot \sqrt[4]{81 x}$

Step 10.6.2

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

$\frac{{\sqrt[4]{x^{3}}}^{3}}{2 x^{8} x^{\frac{3}{4} \cdot 4}} \cdot \sqrt[4]{81 x}$

Step 10.6.3

Combine $\frac{3}{4}$ and $4$ .

$\frac{{\sqrt[4]{x^{3}}}^{3}}{2 x^{8} x^{\frac{3 \cdot 4}{4}}} \cdot \sqrt[4]{81 x}$

Step 10.6.4

Multiply $3$ by $4$ .

$\frac{{\sqrt[4]{x^{3}}}^{3}}{2 x^{8} x^{\frac{12}{4}}} \cdot \sqrt[4]{81 x}$

Step 10.6.5

Cancel the common factor of $12$ and $4$ .

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Step 10.6.5.1

Factor $4$ out of $12$ .

$\frac{{\sqrt[4]{x^{3}}}^{3}}{2 x^{8} x^{\frac{4 \cdot 3}{4}}} \cdot \sqrt[4]{81 x}$

Step 10.6.5.2

Cancel the common factors.

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Step 10.6.5.2.1

Factor $4$ out of $4$ .

$\frac{{\sqrt[4]{x^{3}}}^{3}}{2 x^{8} x^{\frac{4 \cdot 3}{4 (1)}}} \cdot \sqrt[4]{81 x}$

Step 10.6.5.2.2

Cancel the common factor.

$\frac{{\sqrt[4]{x^{3}}}^{3}}{2 x^{8} x^{\frac{4 \cdot 3}{4 \cdot 1}}} \cdot \sqrt[4]{81 x}$

Step 10.6.5.2.3

Rewrite the expression.

$\frac{{\sqrt[4]{x^{3}}}^{3}}{2 x^{8} x^{\frac{3}{1}}} \cdot \sqrt[4]{81 x}$

Step 10.6.5.2.4

Divide $3$ by $1$ .

$\frac{{\sqrt[4]{x^{3}}}^{3}}{2 x^{8} x^{3}} \cdot \sqrt[4]{81 x}$

Step 11

Simplify the numerator.

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Step 11.1

Rewrite ${\sqrt[4]{x^{3}}}^{3}$ as $\sqrt[4]{{(x^{3})}^{3}}$ .

$\frac{\sqrt[4]{{(x^{3})}^{3}}}{2 x^{8} x^{3}} \cdot \sqrt[4]{81 x}$

Step 11.2

Multiply the exponents in ${(x^{3})}^{3}$ .

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Step 11.2.1

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

$\frac{\sqrt[4]{x^{3 \cdot 3}}}{2 x^{8} x^{3}} \cdot \sqrt[4]{81 x}$

Step 11.2.2

Multiply $3$ by $3$ .

$\frac{\sqrt[4]{x^{9}}}{2 x^{8} x^{3}} \cdot \sqrt[4]{81 x}$

Step 11.3

Rewrite $x^{9}$ as ${(x^{2})}^{4} x$ .

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Step 11.3.1

Factor out $x^{8}$ .

$\frac{\sqrt[4]{x^{8} x}}{2 x^{8} x^{3}} \cdot \sqrt[4]{81 x}$

Step 11.3.2

Rewrite $x^{8}$ as ${(x^{2})}^{4}$ .

$\frac{\sqrt[4]{{(x^{2})}^{4} x}}{2 x^{8} x^{3}} \cdot \sqrt[4]{81 x}$

Step 11.4

Pull terms out from under the radical.

$\frac{x^{2} \sqrt[4]{x}}{2 x^{8} x^{3}} \cdot \sqrt[4]{81 x}$

Step 12

Multiply $x^{8}$ by $x^{3}$ by adding the exponents.

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Step 12.1

Move $x^{3}$ .

$\frac{x^{2} \sqrt[4]{x}}{2 (x^{3} x^{8})} \cdot \sqrt[4]{81 x}$

Step 12.2

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

$\frac{x^{2} \sqrt[4]{x}}{2 x^{3 + 8}} \cdot \sqrt[4]{81 x}$

Step 12.3

Add $3$ and $8$ .

$\frac{x^{2} \sqrt[4]{x}}{2 x^{11}} \cdot \sqrt[4]{81 x}$

Step 13

Cancel the common factor of $x^{2}$ and $x^{11}$ .

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Step 13.1

Factor $x^{2}$ out of $x^{2} \sqrt[4]{x}$ .

$\frac{x^{2} (\sqrt[4]{x})}{2 x^{11}} \cdot \sqrt[4]{81 x}$

Step 13.2

Cancel the common factors.

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Step 13.2.1

Factor $x^{2}$ out of $2 x^{11}$ .

$\frac{x^{2} (\sqrt[4]{x})}{x^{2} (2 x^{9})} \cdot \sqrt[4]{81 x}$

Step 13.2.2

Cancel the common factor.

$\frac{x^{2} \sqrt[4]{x}}{x^{2} (2 x^{9})} \cdot \sqrt[4]{81 x}$

Step 13.2.3

Rewrite the expression.

$\frac{\sqrt[4]{x}}{2 x^{9}} \cdot \sqrt[4]{81 x}$

Step 14

Rewrite $81$ as $3^{4}$ .

$\frac{\sqrt[4]{x}}{2 x^{9}} \cdot \sqrt[4]{3^{4} x}$

Step 15

Pull terms out from under the radical.

$\frac{\sqrt[4]{x}}{2 x^{9}} \cdot (3 \sqrt[4]{x})$

Step 16

Rewrite using the commutative property of multiplication.

$3 \frac{\sqrt[4]{x}}{2 x^{9}} \sqrt[4]{x}$

Step 17

Combine $3$ and $\frac{\sqrt[4]{x}}{2 x^{9}}$ .

$\frac{3 \sqrt[4]{x}}{2 x^{9}} \sqrt[4]{x}$

Step 18

Multiply $\frac{3 \sqrt[4]{x}}{2 x^{9}} \sqrt[4]{x}$ .

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Step 18.1

Combine $\frac{3 \sqrt[4]{x}}{2 x^{9}}$ and $\sqrt[4]{x}$ .

$\frac{3 \sqrt[4]{x} \sqrt[4]{x}}{2 x^{9}}$

Step 18.2

Raise $\sqrt[4]{x}$ to the power of $1$ .

$\frac{3 ({\sqrt[4]{x}}^{1} \sqrt[4]{x})}{2 x^{9}}$

Step 18.3

Raise $\sqrt[4]{x}$ to the power of $1$ .

$\frac{3 ({\sqrt[4]{x}}^{1} {\sqrt[4]{x}}^{1})}{2 x^{9}}$

Step 18.4

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

$\frac{3 {\sqrt[4]{x}}^{1 + 1}}{2 x^{9}}$

Step 18.5

Add $1$ and $1$ .

$\frac{3 {\sqrt[4]{x}}^{2}}{2 x^{9}}$

Step 19

Rewrite ${\sqrt[4]{x}}^{2}$ as $\sqrt{x}$ .

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Step 19.1

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

$\frac{3 {(x^{\frac{1}{4}})}^{2}}{2 x^{9}}$

Step 19.2

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

$\frac{3 x^{\frac{1}{4} \cdot 2}}{2 x^{9}}$

Step 19.3

Combine $\frac{1}{4}$ and $2$ .

$\frac{3 x^{\frac{2}{4}}}{2 x^{9}}$

Step 19.4

Cancel the common factor of $2$ and $4$ .

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Step 19.4.1

Factor $2$ out of $2$ .

$\frac{3 x^{\frac{2 (1)}{4}}}{2 x^{9}}$

Step 19.4.2

Cancel the common factors.

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Step 19.4.2.1

Factor $2$ out of $4$ .

$\frac{3 x^{\frac{2 \cdot 1}{2 \cdot 2}}}{2 x^{9}}$

Step 19.4.2.2

Cancel the common factor.

$\frac{3 x^{\frac{2 \cdot 1}{2 \cdot 2}}}{2 x^{9}}$

Step 19.4.2.3

Rewrite the expression.

$\frac{3 x^{\frac{1}{2}}}{2 x^{9}}$

Step 19.5

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

$\frac{3 \sqrt{x}}{2 x^{9}}$