Algebra Examples Radical Expressions and Equations

Simplify

$\sqrt{\frac{7 x}{8}}$

Step 1

Rewrite $\frac{7 x}{8}$ as ${(\frac{1}{2})}^{2} \frac{7 x}{2}$ .

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Step 1.1

Factor the perfect power $1^{2}$ out of $7 x$ .

$\sqrt{\frac{1^{2} (7 x)}{8}}$

Step 1.2

Factor the perfect power $2^{2}$ out of $8$ .

$\sqrt{\frac{1^{2} (7 x)}{2^{2} \cdot 2}}$

Step 1.3

Rearrange the fraction $\frac{1^{2} (7 x)}{2^{2} \cdot 2}$ .

$\sqrt{{(\frac{1}{2})}^{2} \frac{7 x}{2}}$

Step 2

Pull terms out from under the radical.

$\frac{1}{2} \sqrt{\frac{7 x}{2}}$

Step 3

Rewrite $\sqrt{\frac{7 x}{2}}$ as $\frac{\sqrt{7 x}}{\sqrt{2}}$ .

$\frac{1}{2} \cdot \frac{\sqrt{7 x}}{\sqrt{2}}$

Step 4

Multiply $\frac{\sqrt{7 x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{1}{2} (\frac{\sqrt{7 x}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{\sqrt{7 x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{1}{2} \cdot \frac{\sqrt{7 x} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.2

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

$\frac{1}{2} \cdot \frac{\sqrt{7 x} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 5.3

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

$\frac{1}{2} \cdot \frac{\sqrt{7 x} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 5.4

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

$\frac{1}{2} \cdot \frac{\sqrt{7 x} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 5.5

Add $1$ and $1$ .

$\frac{1}{2} \cdot \frac{\sqrt{7 x} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 5.6

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

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Step 5.6.1

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

$\frac{1}{2} \cdot \frac{\sqrt{7 x} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 5.6.2

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

$\frac{1}{2} \cdot \frac{\sqrt{7 x} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 5.6.3

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

$\frac{1}{2} \cdot \frac{\sqrt{7 x} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

$\frac{1}{2} \cdot \frac{\sqrt{7 x} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.6.4.2

Rewrite the expression.

$\frac{1}{2} \cdot \frac{\sqrt{7 x} \sqrt{2}}{2^{1}}$

Step 5.6.5

Evaluate the exponent.

$\frac{1}{2} \cdot \frac{\sqrt{7 x} \sqrt{2}}{2}$

Step 6

Simplify the numerator.

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Step 6.1

Combine using the product rule for radicals.

$\frac{1}{2} \cdot \frac{\sqrt{7 x \cdot 2}}{2}$

Step 6.2

Multiply $2$ by $7$ .

$\frac{1}{2} \cdot \frac{\sqrt{14 x}}{2}$

Step 7

Multiply $\frac{1}{2} \cdot \frac{\sqrt{14 x}}{2}$ .

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Step 7.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{14 x}}{2}$ .

$\frac{\sqrt{14 x}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$\frac{\sqrt{14 x}}{4}$

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