Pre-Algebra Examples

$\frac{\sqrt{x}}{\sqrt{13} + \sqrt{11}}$

Step 1

Multiply $\frac{\sqrt{x}}{\sqrt{13} + \sqrt{11}}$ by $\frac{\sqrt{13} - \sqrt{11}}{\sqrt{13} - \sqrt{11}}$ .

$\frac{\sqrt{x}}{\sqrt{13} + \sqrt{11}} \cdot \frac{\sqrt{13} - \sqrt{11}}{\sqrt{13} - \sqrt{11}}$

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{\sqrt{x}}{\sqrt{13} + \sqrt{11}}$ by $\frac{\sqrt{13} - \sqrt{11}}{\sqrt{13} - \sqrt{11}}$ .

$\frac{\sqrt{x} (\sqrt{13} - \sqrt{11})}{(\sqrt{13} + \sqrt{11}) (\sqrt{13} - \sqrt{11})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{\sqrt{x} (\sqrt{13} - \sqrt{11})}{{\sqrt{13}}^{2} - \sqrt{143} + \sqrt{143} - {\sqrt{11}}^{2}}$

Step 2.3

Simplify.

$\frac{\sqrt{x} (\sqrt{13} - \sqrt{11})}{2}$

Step 2.4

Apply the distributive property.

$\frac{\sqrt{x} \sqrt{13} + \sqrt{x} (- \sqrt{11})}{2}$

Step 2.5

Combine using the product rule for radicals.

$\frac{\sqrt{x \cdot 13} + \sqrt{x} (- \sqrt{11})}{2}$

Step 2.6

Combine using the product rule for radicals.

$\frac{\sqrt{x \cdot 13} - \sqrt{x \cdot 11}}{2}$

Step 2.7

Reorder factors in $\sqrt{x \cdot 13} - \sqrt{x \cdot 11}$ .

$\frac{\sqrt{13 x} - \sqrt{11 x}}{2}$

Step 3

Simplify the numerator.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13 x}$ as ${(13 x)}^{\frac{1}{2}}$ .

$\frac{{(13 x)}^{\frac{1}{2}} - \sqrt{11 x}}{2}$

Step 3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 x}$ as ${(11 x)}^{\frac{1}{2}}$ .

$\frac{{(13 x)}^{\frac{1}{2}} - {(11 x)}^{\frac{1}{2}}}{2}$

Step 3.3

Apply the product rule to $13 x$ .

$\frac{13^{\frac{1}{2}} x^{\frac{1}{2}} - {(11 x)}^{\frac{1}{2}}}{2}$

Step 3.4

Apply the product rule to $11 x$ .

$\frac{13^{\frac{1}{2}} x^{\frac{1}{2}} - (11^{\frac{1}{2}} x^{\frac{1}{2}})}{2}$

Step 3.5

Factor $x^{\frac{1}{2}}$ out of $13^{\frac{1}{2}} x^{\frac{1}{2}} - 11^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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Step 3.5.1

Factor $x^{\frac{1}{2}}$ out of $13^{\frac{1}{2}} x^{\frac{1}{2}}$ .

$\frac{x^{\frac{1}{2}} \cdot 13^{\frac{1}{2}} - 11^{\frac{1}{2}} x^{\frac{1}{2}}}{2}$

Step 3.5.2

Factor $x^{\frac{1}{2}}$ out of $- 11^{\frac{1}{2}} x^{\frac{1}{2}}$ .

$\frac{x^{\frac{1}{2}} \cdot 13^{\frac{1}{2}} + x^{\frac{1}{2}} (- 11^{\frac{1}{2}})}{2}$

Step 3.5.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \cdot 13^{\frac{1}{2}} + x^{\frac{1}{2}} (- 11^{\frac{1}{2}})$ .

$\frac{x^{\frac{1}{2}} (13^{\frac{1}{2}} - 11^{\frac{1}{2}})}{2}$