Algebra Examples

$\sqrt{\frac{3 x}{2}} \cdot \sqrt{\frac{x}{6}}$

Step 1

Rewrite $\sqrt{\frac{3 x}{2}}$ as $\frac{\sqrt{3 x}}{\sqrt{2}}$ .

$\frac{\sqrt{3 x}}{\sqrt{2}} \cdot \sqrt{\frac{x}{6}}$

Step 2

Multiply $\frac{\sqrt{3 x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{3 x}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \cdot \sqrt{\frac{x}{6}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt{3 x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{3 x} \sqrt{2}}{\sqrt{2} \sqrt{2}} \cdot \sqrt{\frac{x}{6}}$

Step 3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{3 x} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} \cdot \sqrt{\frac{x}{6}}$

Step 3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{3 x} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} \cdot \sqrt{\frac{x}{6}}$

Step 3.4

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

$\frac{\sqrt{3 x} \sqrt{2}}{{\sqrt{2}}^{1 + 1}} \cdot \sqrt{\frac{x}{6}}$

Step 3.5

Add $1$ and $1$ .

$\frac{\sqrt{3 x} \sqrt{2}}{{\sqrt{2}}^{2}} \cdot \sqrt{\frac{x}{6}}$

Step 3.6

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

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Step 3.6.1

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

$\frac{\sqrt{3 x} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} \cdot \sqrt{\frac{x}{6}}$

Step 3.6.2

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

$\frac{\sqrt{3 x} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} \cdot \sqrt{\frac{x}{6}}$

Step 3.6.3

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

$\frac{\sqrt{3 x} \sqrt{2}}{2^{\frac{2}{2}}} \cdot \sqrt{\frac{x}{6}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{\sqrt{3 x} \sqrt{2}}{2^{\frac{2}{2}}} \cdot \sqrt{\frac{x}{6}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{\sqrt{3 x} \sqrt{2}}{2^{1}} \cdot \sqrt{\frac{x}{6}}$

Step 3.6.5

Evaluate the exponent.

$\frac{\sqrt{3 x} \sqrt{2}}{2} \cdot \sqrt{\frac{x}{6}}$

Step 4

Simplify the numerator.

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Step 4.1

Combine using the product rule for radicals.

$\frac{\sqrt{3 x \cdot 2}}{2} \cdot \sqrt{\frac{x}{6}}$

Step 4.2

Multiply $2$ by $3$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 7.6

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

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Step 7.6.1

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

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

Step 7.6.2

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

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

Step 7.6.3

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

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

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

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

Step 7.6.5

Evaluate the exponent.

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

Step 8

Combine using the product rule for radicals.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

Combine using the product rule for radicals.

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

Step 9.3

Multiply $6$ by $6$ .

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

Step 9.4

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

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

Step 9.5

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

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

Step 9.6

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

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

Step 9.7

Add $1$ and $1$ .

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

Step 9.8

Multiply $2$ by $6$ .

$\frac{\sqrt{36 x^{2}}}{12}$

Step 10

Simplify the numerator.

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Step 10.1

Rewrite $36 x^{2}$ as ${(6 x)}^{2}$ .

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

Step 10.2

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

$\frac{6 x}{12}$

Step 11

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

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Step 11.1

Factor $6$ out of $6 x$ .

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

Step 11.2

Cancel the common factors.

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Step 11.2.1

Factor $6$ out of $12$ .

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

Step 11.2.2

Cancel the common factor.

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

Step 11.2.3

Rewrite the expression.

$\frac{x}{2}$