Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

Reorder $3$ and $x^{2}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

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

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

Step 4

Expand the denominator using the FOIL method.

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

Step 5

Simplify.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Cancel the common factor of $2$ .

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Step 5.1.4.1

Cancel the common factor.

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

Step 5.1.4.2

Rewrite the expression.

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

Step 5.1.5

Evaluate the exponent.

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

Step 5.2

Multiply $3$ by $- 1$ .

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

Step 6

Factor $3$ out of $9 - 3 x^{2}$ .

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Step 6.1

Factor $3$ out of $9$ .

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

Step 6.2

Factor $3$ out of $- 3 x^{2}$ .

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

Step 6.3

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

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