Simplify square root of x/(8y)

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

Step 1

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

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

$\sqrt{\frac{1^{2} x}{8 y}}$

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

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

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 6

Combine using the product rule for radicals.

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

Step 7

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

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

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

Multiply $2$ by $2$ .

$\frac{\sqrt{x \cdot 2 y}}{4 y}$

Step 8

Reorder factors in $\frac{\sqrt{x \cdot 2 y}}{4 y}$ .

$\frac{\sqrt{2 x y}}{4 y}$

Simplify square root of x/(8y)

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