Evaluate ( square root of 11+ square root of 5)/( square root of 11- square root of 5)

$\frac{\sqrt{11} + \sqrt{5}}{\sqrt{11} - \sqrt{5}}$

Step 1

Multiply $\frac{\sqrt{11} + \sqrt{5}}{\sqrt{11} - \sqrt{5}}$ by $\frac{\sqrt{11} + \sqrt{5}}{\sqrt{11} + \sqrt{5}}$ .

$\frac{\sqrt{11} + \sqrt{5}}{\sqrt{11} - \sqrt{5}} \cdot \frac{\sqrt{11} + \sqrt{5}}{\sqrt{11} + \sqrt{5}}$

Step 2

Combine fractions.

Multiply $\frac{\sqrt{11} + \sqrt{5}}{\sqrt{11} - \sqrt{5}}$ by $\frac{\sqrt{11} + \sqrt{5}}{\sqrt{11} + \sqrt{5}}$ .

$\frac{(\sqrt{11} + \sqrt{5}) (\sqrt{11} + \sqrt{5})}{(\sqrt{11} - \sqrt{5}) (\sqrt{11} + \sqrt{5})}$

Expand the denominator using the FOIL method.

$\frac{(\sqrt{11} + \sqrt{5}) (\sqrt{11} + \sqrt{5})}{{\sqrt{11}}^{2} + \sqrt{55} - \sqrt{55} - {\sqrt{5}}^{2}}$

Simplify.

$\frac{(\sqrt{11} + \sqrt{5}) (\sqrt{11} + \sqrt{5})}{6}$

Step 3

Simplify the numerator.

Raise $\sqrt{11} + \sqrt{5}$ to the power of $1$ .

$\frac{{(\sqrt{11} + \sqrt{5})}^{1} (\sqrt{11} + \sqrt{5})}{6}$

Raise $\sqrt{11} + \sqrt{5}$ to the power of $1$ .

$\frac{{(\sqrt{11} + \sqrt{5})}^{1} {(\sqrt{11} + \sqrt{5})}^{1}}{6}$

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

$\frac{{(\sqrt{11} + \sqrt{5})}^{1 + 1}}{6}$

Add $1$ and $1$ .

$\frac{{(\sqrt{11} + \sqrt{5})}^{2}}{6}$

Step 4

Rewrite ${(\sqrt{11} + \sqrt{5})}^{2}$ as $(\sqrt{11} + \sqrt{5}) (\sqrt{11} + \sqrt{5})$ .

$\frac{(\sqrt{11} + \sqrt{5}) (\sqrt{11} + \sqrt{5})}{6}$

Step 5

Expand $(\sqrt{11} + \sqrt{5}) (\sqrt{11} + \sqrt{5})$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{11} (\sqrt{11} + \sqrt{5}) + \sqrt{5} (\sqrt{11} + \sqrt{5})}{6}$

Apply the distributive property.

$\frac{\sqrt{11} \sqrt{11} + \sqrt{11} \sqrt{5} + \sqrt{5} (\sqrt{11} + \sqrt{5})}{6}$

Apply the distributive property.

$\frac{\sqrt{11} \sqrt{11} + \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

Step 6

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\frac{\sqrt{11 \cdot 11} + \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

Multiply $11$ by $11$ .

$\frac{\sqrt{121} + \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

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

$\frac{\sqrt{11^{2}} + \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

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

$\frac{11 + \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

Combine using the product rule for radicals.

$\frac{11 + \sqrt{11 \cdot 5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

Multiply $11$ by $5$ .

$\frac{11 + \sqrt{55} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

Combine using the product rule for radicals.

$\frac{11 + \sqrt{55} + \sqrt{5 \cdot 11} + \sqrt{5} \sqrt{5}}{6}$

Multiply $5$ by $11$ .

$\frac{11 + \sqrt{55} + \sqrt{55} + \sqrt{5} \sqrt{5}}{6}$

Combine using the product rule for radicals.

$\frac{11 + \sqrt{55} + \sqrt{55} + \sqrt{5 \cdot 5}}{6}$

Multiply $5$ by $5$ .

$\frac{11 + \sqrt{55} + \sqrt{55} + \sqrt{25}}{6}$

Rewrite $25$ as $5^{2}$ .

$\frac{11 + \sqrt{55} + \sqrt{55} + \sqrt{5^{2}}}{6}$

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

$\frac{11 + \sqrt{55} + \sqrt{55} + 5}{6}$

Add $11$ and $5$ .

$\frac{16 + \sqrt{55} + \sqrt{55}}{6}$

Add $\sqrt{55}$ and $\sqrt{55}$ .

$\frac{16 + 2 \sqrt{55}}{6}$

Step 7

Cancel the common factor of $16 + 2 \sqrt{55}$ and $6$ .

Factor $2$ out of $16$ .

$\frac{2 \cdot 8 + 2 \sqrt{55}}{6}$

Factor $2$ out of $2 \sqrt{55}$ .

$\frac{2 \cdot 8 + 2 (\sqrt{55})}{6}$

Factor $2$ out of $2 \cdot 8 + 2 (\sqrt{55})$ .

$\frac{2 \cdot (8 + \sqrt{55})}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$\frac{2 \cdot (8 + \sqrt{55})}{2 (3)}$

Cancel the common factor.

$\frac{2 \cdot (8 + \sqrt{55})}{2 \cdot 3}$

Rewrite the expression.

$\frac{8 + \sqrt{55}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{8 + \sqrt{55}}{3}$

Decimal Form:

$5.13873282 \dots$

Evaluate ( square root of 11+ square root of 5)/( square root of 11- square root of 5)

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