Algebra Examples

$\sqrt{\frac{6}{x}} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 1

Rewrite $\sqrt{\frac{6}{x}}$ as $\frac{\sqrt{6}}{\sqrt{x}}$ .

$\frac{\sqrt{6}}{\sqrt{x}} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 2

Multiply $\frac{\sqrt{6}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{\sqrt{6}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt{6}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{\sqrt{6} \sqrt{x}}{\sqrt{x} \sqrt{x}} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 3.2

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

$\frac{\sqrt{6} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 3.3

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

$\frac{\sqrt{6} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 3.4

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

$\frac{\sqrt{6} \sqrt{x}}{{\sqrt{x}}^{1 + 1}} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 3.5

Add $1$ and $1$ .

$\frac{\sqrt{6} \sqrt{x}}{{\sqrt{x}}^{2}} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 3.6

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

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Step 3.6.1

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

$\frac{\sqrt{6} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 3.6.2

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

$\frac{\sqrt{6} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 3.6.3

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

$\frac{\sqrt{6} \sqrt{x}}{x^{\frac{2}{2}}} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{\sqrt{6} \sqrt{x}}{x^{\frac{2}{2}}} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{\sqrt{6} \sqrt{x}}{x^{1}} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 3.6.5

Simplify.

$\frac{\sqrt{6} \sqrt{x}}{x} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 4

Combine using the product rule for radicals.

$\frac{\sqrt{6 x}}{x} \cdot \sqrt{\frac{x^{2}}{24}}$

Step 5

Rewrite $\frac{x^{2}}{24}$ as ${(\frac{x}{2})}^{2} \frac{1}{6}$ .

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Step 5.1

Factor the perfect power $x^{2}$ out of $x^{2}$ .

$\frac{\sqrt{6 x}}{x} \cdot \sqrt{\frac{x^{2} \cdot 1}{24}}$

Step 5.2

Factor the perfect power $2^{2}$ out of $24$ .

$\frac{\sqrt{6 x}}{x} \cdot \sqrt{\frac{x^{2} \cdot 1}{2^{2} \cdot 6}}$

Step 5.3

Rearrange the fraction $\frac{x^{2} \cdot 1}{2^{2} \cdot 6}$ .

$\frac{\sqrt{6 x}}{x} \cdot \sqrt{{(\frac{x}{2})}^{2} \frac{1}{6}}$

Step 6

Pull terms out from under the radical.

$\frac{\sqrt{6 x}}{x} \cdot (\frac{x}{2} \sqrt{\frac{1}{6}})$

Step 7

Rewrite using the commutative property of multiplication.

$\sqrt{\frac{1}{6}} \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 8

Rewrite $\sqrt{\frac{1}{6}}$ as $\frac{\sqrt{1}}{\sqrt{6}}$ .

$\frac{\sqrt{1}}{\sqrt{6}} \cdot \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 9

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 10

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 11

Combine and simplify the denominator.

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Step 11.1

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{6}}{\sqrt{6} \sqrt{6}} \cdot \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 11.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}} \cdot \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 11.3

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}} \cdot \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 11.4

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

$\frac{\sqrt{6}}{{\sqrt{6}}^{1 + 1}} \cdot \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 11.5

Add $1$ and $1$ .

$\frac{\sqrt{6}}{{\sqrt{6}}^{2}} \cdot \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 11.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 11.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{\sqrt{6}}{{(6^{\frac{1}{2}})}^{2}} \cdot \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 11.6.2

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

$\frac{\sqrt{6}}{6^{\frac{1}{2} \cdot 2}} \cdot \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 11.6.3

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

$\frac{\sqrt{6}}{6^{\frac{2}{2}}} \cdot \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 11.6.4

Cancel the common factor of $2$ .

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Step 11.6.4.1

Cancel the common factor.

$\frac{\sqrt{6}}{6^{\frac{2}{2}}} \cdot \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 11.6.4.2

Rewrite the expression.

$\frac{\sqrt{6}}{6^{1}} \cdot \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 11.6.5

Evaluate the exponent.

$\frac{\sqrt{6}}{6} \cdot \frac{\sqrt{6 x}}{x} \frac{x}{2}$

Step 12

Multiply $\frac{\sqrt{6}}{6} \cdot \frac{\sqrt{6 x}}{x}$ .

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Step 12.1

Multiply $\frac{\sqrt{6}}{6}$ by $\frac{\sqrt{6 x}}{x}$ .

$\frac{\sqrt{6} \sqrt{6 x}}{6 x} \cdot \frac{x}{2}$

Step 12.2

Combine using the product rule for radicals.

$\frac{\sqrt{6 (6 x)}}{6 x} \cdot \frac{x}{2}$

Step 12.3

Multiply $6$ by $6$ .

$\frac{\sqrt{36 x}}{6 x} \cdot \frac{x}{2}$

Step 13

Simplify terms.

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Step 13.1

Cancel the common factor of $x$ .

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Step 13.1.1

Factor $x$ out of $6 x$ .

$\frac{\sqrt{36 x}}{x \cdot 6} \cdot \frac{x}{2}$

Step 13.1.2

Cancel the common factor.

$\frac{\sqrt{36 x}}{x \cdot 6} \cdot \frac{x}{2}$

Step 13.1.3

Rewrite the expression.

$\frac{\sqrt{36 x}}{6} \cdot \frac{1}{2}$

Step 13.2

Multiply $\frac{\sqrt{36 x}}{6}$ by $\frac{1}{2}$ .

$\frac{\sqrt{36 x}}{6 \cdot 2}$

Step 13.3

Multiply $6$ by $2$ .

$\frac{\sqrt{36 x}}{12}$

Step 14

Simplify the numerator.

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Step 14.1

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{6^{2} x}}{12}$

Step 14.2

Pull terms out from under the radical.

$\frac{6 \sqrt{x}}{12}$

Step 15

Cancel the common factor of $6$ and $12$ .

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Step 15.1

Factor $6$ out of $6 \sqrt{x}$ .

$\frac{6 (\sqrt{x})}{12}$

Step 15.2

Cancel the common factors.

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Step 15.2.1

Factor $6$ out of $12$ .

$\frac{6 \sqrt{x}}{6 \cdot 2}$

Step 15.2.2

Cancel the common factor.

$\frac{6 \sqrt{x}}{6 \cdot 2}$

Step 15.2.3

Rewrite the expression.

$\frac{\sqrt{x}}{2}$