Simplify ( square root of 2x)/(3- square root of 2x)

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

Step 1

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

$\frac{\sqrt{2 x}}{3 - \sqrt{2 x}} \cdot \frac{3 + \sqrt{2 x}}{3 + \sqrt{2 x}}$

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{2 x} (3 + \sqrt{2 x})}{9 + 3 \sqrt{2 x} - 3 \sqrt{2 x} - {\sqrt{2 x}}^{2}}$

Step 4

Simplify.

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

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

$\frac{\sqrt{2 x} (3 + \sqrt{2 x})}{9 - {({(2 x)}^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt{2 x} (3 + \sqrt{2 x})}{9 - {(2 x)}^{\frac{1}{2} \cdot 2}}$

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

$\frac{\sqrt{2 x} (3 + \sqrt{2 x})}{9 - {(2 x)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{2 x} (3 + \sqrt{2 x})}{9 - {(2 x)}^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{\sqrt{2 x} (3 + \sqrt{2 x})}{9 - {(2 x)}^{1}}$

Simplify.

$\frac{\sqrt{2 x} (3 + \sqrt{2 x})}{9 - (2 x)}$

Multiply $2$ by $- 1$ .

$\frac{\sqrt{2 x} (3 + \sqrt{2 x})}{9 - 2 x}$

Step 5

Apply the distributive property.

$\frac{\sqrt{2 x} \cdot 3 + \sqrt{2 x} \sqrt{2 x}}{9 - 2 x}$

Step 6

Move $3$ to the left of $\sqrt{2 x}$ .

$\frac{3 \cdot \sqrt{2 x} + \sqrt{2 x} \sqrt{2 x}}{9 - 2 x}$

Step 7

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

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

$\frac{3 \cdot \sqrt{2 x} + {\sqrt{2 x}}^{1} \sqrt{2 x}}{9 - 2 x}$

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

$\frac{3 \cdot \sqrt{2 x} + {\sqrt{2 x}}^{1} {\sqrt{2 x}}^{1}}{9 - 2 x}$

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

$\frac{3 \cdot \sqrt{2 x} + {\sqrt{2 x}}^{1 + 1}}{9 - 2 x}$

Add $1$ and $1$ .

$\frac{3 \cdot \sqrt{2 x} + {\sqrt{2 x}}^{2}}{9 - 2 x}$

Step 8

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

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

$\frac{3 \sqrt{2 x} + {({(2 x)}^{\frac{1}{2}})}^{2}}{9 - 2 x}$

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

$\frac{3 \sqrt{2 x} + {(2 x)}^{\frac{1}{2} \cdot 2}}{9 - 2 x}$

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

$\frac{3 \sqrt{2 x} + {(2 x)}^{\frac{2}{2}}}{9 - 2 x}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 \sqrt{2 x} + {(2 x)}^{\frac{2}{2}}}{9 - 2 x}$

Rewrite the expression.

$\frac{3 \sqrt{2 x} + {(2 x)}^{1}}{9 - 2 x}$

Simplify.

$\frac{3 \sqrt{2 x} + 2 x}{9 - 2 x}$

Simplify ( square root of 2x)/(3- square root of 2x)

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