Precalculus Examples

$\frac{x^{2} - 121}{\sqrt{x} - \sqrt{11}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $121$ as $11^{2}$ .

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

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 11$ .

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Expand the denominator using the FOIL method.

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

Step 3.3

Simplify.

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

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

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

Step 3.3.1.3

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

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

Step 3.3.1.4

Cancel the common factor of $2$ .

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Step 3.3.1.4.1

Cancel the common factor.

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

Step 3.3.1.4.2

Rewrite the expression.

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

Step 3.3.1.5

Simplify.

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

Step 3.3.2

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 3.3.2.1

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

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

Step 3.3.2.2

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

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

Step 3.3.2.3

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

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

Step 3.3.2.4

Cancel the common factor of $2$ .

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Step 3.3.2.4.1

Cancel the common factor.

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

Step 3.3.2.4.2

Rewrite the expression.

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

Step 3.3.2.5

Evaluate the exponent.

$\frac{(x + 11) (x - 11) (\sqrt{x} + \sqrt{11})}{x - 1 \cdot 11}$

Step 3.3.3

Multiply $- 1$ by $11$ .

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

Step 3.4

Cancel the common factor of $x - 11$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Divide $(x + 11) (\sqrt{x} + \sqrt{11})$ by $1$ .

$(x + 11) (\sqrt{x} + \sqrt{11})$

Step 4

Expand $(x + 11) (\sqrt{x} + \sqrt{11})$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$x (\sqrt{x} + \sqrt{11}) + 11 (\sqrt{x} + \sqrt{11})$

Step 4.2

Apply the distributive property.

$x \sqrt{x} + x \sqrt{11} + 11 (\sqrt{x} + \sqrt{11})$

Step 4.3

Apply the distributive property.

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