Simplify ( square root of 8x)/( square root of 3y)

$\frac{\sqrt{8 x}}{\sqrt{3 y}}$

Step 1

Simplify the numerator.

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

Factor $4$ out of $8$ .

$\frac{\sqrt{4 (2) x}}{\sqrt{3 y}}$

Rewrite $4$ as $2^{2}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

$\frac{2 \sqrt{2 x}}{\sqrt{3 y}}$

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

$\frac{2 \sqrt{2 x} \sqrt{3 y}}{\sqrt{3 y} \sqrt{3 y}}$

Raise $\sqrt{3 y}$ to the power of $1$ .

$\frac{2 \sqrt{2 x} \sqrt{3 y}}{{\sqrt{3 y}}^{1} \sqrt{3 y}}$

Raise $\sqrt{3 y}$ to the power of $1$ .

$\frac{2 \sqrt{2 x} \sqrt{3 y}}{{\sqrt{3 y}}^{1} {\sqrt{3 y}}^{1}}$

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3 y}}^{2}$ as $3 y$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 y}$ as ${(3 y)}^{\frac{1}{2}}$ .

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

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

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

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

$\frac{2 \sqrt{2 x} \sqrt{3 y}}{{(3 y)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{2 x} \sqrt{3 y}}{{(3 y)}^{\frac{2}{2}}}$

Rewrite the expression.

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

Simplify.

$\frac{2 \sqrt{2 x} \sqrt{3 y}}{3 y}$

Step 4

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{2 \sqrt{3 y (2 x)}}{3 y}$

Multiply $2$ by $3$ .

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

Simplify ( square root of 8x)/( square root of 3y)

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