Precalculus Examples

$\frac{\sqrt{11} + \sqrt{2}}{\sqrt{11} - \sqrt{2}}$

Step 1

Multiply $\frac{\sqrt{11} + \sqrt{2}}{\sqrt{11} - \sqrt{2}}$ by $\frac{\sqrt{11} + \sqrt{2}}{\sqrt{11} + \sqrt{2}}$ .

$\frac{\sqrt{11} + \sqrt{2}}{\sqrt{11} - \sqrt{2}} \cdot \frac{\sqrt{11} + \sqrt{2}}{\sqrt{11} + \sqrt{2}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{11} + \sqrt{2}}{\sqrt{11} - \sqrt{2}}$ by $\frac{\sqrt{11} + \sqrt{2}}{\sqrt{11} + \sqrt{2}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{11} + \sqrt{2}) (\sqrt{11} + \sqrt{2})}{{\sqrt{11}}^{2} + \sqrt{22} - \sqrt{22} - {\sqrt{2}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{11} + \sqrt{2}) (\sqrt{11} + \sqrt{2})}{9}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $\sqrt{11} + \sqrt{2}$ to the power of $1$ .

$\frac{{(\sqrt{11} + \sqrt{2})}^{1} (\sqrt{11} + \sqrt{2})}{9}$

Step 3.2

Raise $\sqrt{11} + \sqrt{2}$ to the power of $1$ .

$\frac{{(\sqrt{11} + \sqrt{2})}^{1} {(\sqrt{11} + \sqrt{2})}^{1}}{9}$

Step 3.3

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

$\frac{{(\sqrt{11} + \sqrt{2})}^{1 + 1}}{9}$

Step 3.4

Add $1$ and $1$ .

$\frac{{(\sqrt{11} + \sqrt{2})}^{2}}{9}$

Step 4

Rewrite ${(\sqrt{11} + \sqrt{2})}^{2}$ as $(\sqrt{11} + \sqrt{2}) (\sqrt{11} + \sqrt{2})$ .

$\frac{(\sqrt{11} + \sqrt{2}) (\sqrt{11} + \sqrt{2})}{9}$

Step 5

Expand $(\sqrt{11} + \sqrt{2}) (\sqrt{11} + \sqrt{2})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{\sqrt{11} (\sqrt{11} + \sqrt{2}) + \sqrt{2} (\sqrt{11} + \sqrt{2})}{9}$

Step 5.2

Apply the distributive property.

$\frac{\sqrt{11} \sqrt{11} + \sqrt{11} \sqrt{2} + \sqrt{2} (\sqrt{11} + \sqrt{2})}{9}$

Step 5.3

Apply the distributive property.

$\frac{\sqrt{11} \sqrt{11} + \sqrt{11} \sqrt{2} + \sqrt{2} \sqrt{11} + \sqrt{2} \sqrt{2}}{9}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{11 \cdot 11} + \sqrt{11} \sqrt{2} + \sqrt{2} \sqrt{11} + \sqrt{2} \sqrt{2}}{9}$

Step 6.1.2

Multiply $11$ by $11$ .

$\frac{\sqrt{121} + \sqrt{11} \sqrt{2} + \sqrt{2} \sqrt{11} + \sqrt{2} \sqrt{2}}{9}$

Step 6.1.3

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

$\frac{\sqrt{11^{2}} + \sqrt{11} \sqrt{2} + \sqrt{2} \sqrt{11} + \sqrt{2} \sqrt{2}}{9}$

Step 6.1.4

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

$\frac{11 + \sqrt{11} \sqrt{2} + \sqrt{2} \sqrt{11} + \sqrt{2} \sqrt{2}}{9}$

Step 6.1.5

Combine using the product rule for radicals.

$\frac{11 + \sqrt{11 \cdot 2} + \sqrt{2} \sqrt{11} + \sqrt{2} \sqrt{2}}{9}$

Step 6.1.6

Multiply $11$ by $2$ .

$\frac{11 + \sqrt{22} + \sqrt{2} \sqrt{11} + \sqrt{2} \sqrt{2}}{9}$

Step 6.1.7

Combine using the product rule for radicals.

$\frac{11 + \sqrt{22} + \sqrt{2 \cdot 11} + \sqrt{2} \sqrt{2}}{9}$

Step 6.1.8

Multiply $2$ by $11$ .

$\frac{11 + \sqrt{22} + \sqrt{22} + \sqrt{2} \sqrt{2}}{9}$

Step 6.1.9

Combine using the product rule for radicals.

$\frac{11 + \sqrt{22} + \sqrt{22} + \sqrt{2 \cdot 2}}{9}$

Step 6.1.10

Multiply $2$ by $2$ .

$\frac{11 + \sqrt{22} + \sqrt{22} + \sqrt{4}}{9}$

Step 6.1.11

Rewrite $4$ as $2^{2}$ .

$\frac{11 + \sqrt{22} + \sqrt{22} + \sqrt{2^{2}}}{9}$

Step 6.1.12

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

$\frac{11 + \sqrt{22} + \sqrt{22} + 2}{9}$

Step 6.2

Add $11$ and $2$ .

$\frac{13 + \sqrt{22} + \sqrt{22}}{9}$

Step 6.3

Add $\sqrt{22}$ and $\sqrt{22}$ .

$\frac{13 + 2 \sqrt{22}}{9}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{13 + 2 \sqrt{22}}{9}$

Decimal Form:

$2.48675905 \dots$