Algebra Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.1.3

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

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

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Cancel the common factor.

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

Step 2.3.1.4.2

Rewrite the expression.

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

Step 2.3.1.5

Evaluate the exponent.

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

Step 2.3.2

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

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.2.4

Cancel the common factor of $2$ .

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Step 2.3.2.4.1

Cancel the common factor.

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

Step 2.3.2.4.2

Rewrite the expression.

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

Step 2.3.2.5

Simplify.

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

Step 2.4

Apply the distributive property.

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

Step 2.5

Combine using the product rule for radicals.

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

Step 2.6

Combine using the product rule for radicals.

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

Step 3

Simplify each term.

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Step 3.1

Multiply $3$ by $3$ .

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

Step 3.2

Rewrite $9$ as $3^{2}$ .

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

Step 3.3

Pull terms out from under the radical, assuming positive real numbers.

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