Simplify ( square root of 8)/( square root of 2x+6)

$\frac{\sqrt{8}}{\sqrt{2 x} + 6}$

Step 1

Simplify the numerator.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$\frac{\sqrt{4 (2)}}{\sqrt{2 x} + 6}$

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

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

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

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

$\frac{2 \sqrt{2} (\sqrt{2 x} - 6)}{(\sqrt{2 x} + 6) (\sqrt{2 x} - 6)}$

Step 4

Expand the denominator using the FOIL method.

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

Step 5

Simplify.

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$\frac{2 \sqrt{2} (\sqrt{2 x} - 6)}{2 x - 36}$

Step 6

Cancel the common factor of $2$ and $2 x - 36$ .

Factor $2$ out of $2 \sqrt{2} (\sqrt{2 x} - 6)$ .

$\frac{2 (\sqrt{2} (\sqrt{2 x} - 6))}{2 x - 36}$

Cancel the common factors.

Factor $2$ out of $2 x$ .

$\frac{2 (\sqrt{2} (\sqrt{2 x} - 6))}{2 (x) - 36}$

Factor $2$ out of $- 36$ .

$\frac{2 (\sqrt{2} (\sqrt{2 x} - 6))}{2 x + 2 \cdot - 18}$

Factor $2$ out of $2 x + 2 \cdot - 18$ .

$\frac{2 (\sqrt{2} (\sqrt{2 x} - 6))}{2 (x - 18)}$

Cancel the common factor.

$\frac{2 (\sqrt{2} (\sqrt{2 x} - 6))}{2 (x - 18)}$

Rewrite the expression.

$\frac{\sqrt{2} (\sqrt{2 x} - 6)}{x - 18}$

Step 7

Apply the distributive property.

$\frac{\sqrt{2} \sqrt{2 x} + \sqrt{2} \cdot - 6}{x - 18}$

Step 8

Multiply $\sqrt{2} \sqrt{2 x}$ .

Combine using the product rule for radicals.

$\frac{\sqrt{2 (2 x)} + \sqrt{2} \cdot - 6}{x - 18}$

Multiply $2$ by $2$ .

$\frac{\sqrt{4 x} + \sqrt{2} \cdot - 6}{x - 18}$

Step 9

Move $- 6$ to the left of $\sqrt{2}$ .

$\frac{\sqrt{4 x} - 6 \cdot \sqrt{2}}{x - 18}$

Step 10

Simplify each term.

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

$\frac{\sqrt{2^{2} x} - 6 \cdot \sqrt{2}}{x - 18}$

Pull terms out from under the radical.

$\frac{2 \sqrt{x} - 6 \sqrt{2}}{x - 18}$

Step 11

Factor $2$ out of $2 \sqrt{x} - 6 \sqrt{2}$ .

Factor $2$ out of $2 \sqrt{x}$ .

$\frac{2 (\sqrt{x}) - 6 \sqrt{2}}{x - 18}$

Factor $2$ out of $- 6 \sqrt{2}$ .

$\frac{2 (\sqrt{x}) + 2 (- 3 \sqrt{2})}{x - 18}$

Factor $2$ out of $2 (\sqrt{x}) + 2 (- 3 \sqrt{2})$ .

$\frac{2 (\sqrt{x} - 3 \sqrt{2})}{x - 18}$

Simplify ( square root of 8)/( square root of 2x+6)

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