Evaluate (3 square root of 11+ square root of 5)/( square root of 11- square root of 5)

$\frac{3 \sqrt{11} + \sqrt{5}}{\sqrt{11} - \sqrt{5}}$

Step 1

Multiply $\frac{3 \sqrt{11} + \sqrt{5}}{\sqrt{11} - \sqrt{5}}$ by $\frac{\sqrt{11} + \sqrt{5}}{\sqrt{11} + \sqrt{5}}$ .

$\frac{3 \sqrt{11} + \sqrt{5}}{\sqrt{11} - \sqrt{5}} \cdot \frac{\sqrt{11} + \sqrt{5}}{\sqrt{11} + \sqrt{5}}$

Step 2

Combine fractions.

Multiply $\frac{3 \sqrt{11} + \sqrt{5}}{\sqrt{11} - \sqrt{5}}$ by $\frac{\sqrt{11} + \sqrt{5}}{\sqrt{11} + \sqrt{5}}$ .

$\frac{(3 \sqrt{11} + \sqrt{5}) (\sqrt{11} + \sqrt{5})}{(\sqrt{11} - \sqrt{5}) (\sqrt{11} + \sqrt{5})}$

Expand the denominator using the FOIL method.

$\frac{(3 \sqrt{11} + \sqrt{5}) (\sqrt{11} + \sqrt{5})}{{\sqrt{11}}^{2} + \sqrt{55} - \sqrt{55} - {\sqrt{5}}^{2}}$

Simplify.

$\frac{(3 \sqrt{11} + \sqrt{5}) (\sqrt{11} + \sqrt{5})}{6}$

Step 3

Expand $(3 \sqrt{11} + \sqrt{5}) (\sqrt{11} + \sqrt{5})$ using the FOIL Method.

Apply the distributive property.

$\frac{3 \sqrt{11} (\sqrt{11} + \sqrt{5}) + \sqrt{5} (\sqrt{11} + \sqrt{5})}{6}$

Apply the distributive property.

$\frac{3 \sqrt{11} \sqrt{11} + 3 \sqrt{11} \sqrt{5} + \sqrt{5} (\sqrt{11} + \sqrt{5})}{6}$

Apply the distributive property.

$\frac{3 \sqrt{11} \sqrt{11} + 3 \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $3 \sqrt{11} \sqrt{11}$ .

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

$\frac{3 ({\sqrt{11}}^{1} \sqrt{11}) + 3 \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

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

$\frac{3 ({\sqrt{11}}^{1} {\sqrt{11}}^{1}) + 3 \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

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

$\frac{3 {\sqrt{11}}^{1 + 1} + 3 \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

Add $1$ and $1$ .

$\frac{3 {\sqrt{11}}^{2} + 3 \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

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

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

$\frac{3 {(11^{\frac{1}{2}})}^{2} + 3 \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

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

$\frac{3 \cdot 11^{\frac{1}{2} \cdot 2} + 3 \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

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

$\frac{3 \cdot 11^{\frac{2}{2}} + 3 \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 \cdot 11^{\frac{2}{2}} + 3 \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

Rewrite the expression.

$\frac{3 \cdot 11^{1} + 3 \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

Evaluate the exponent.

$\frac{3 \cdot 11 + 3 \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

Multiply $3$ by $11$ .

$\frac{33 + 3 \sqrt{11} \sqrt{5} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

Multiply $3 \sqrt{11} \sqrt{5}$ .

Combine using the product rule for radicals.

$\frac{33 + 3 \sqrt{5 \cdot 11} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

Multiply $5$ by $11$ .

$\frac{33 + 3 \sqrt{55} + \sqrt{5} \sqrt{11} + \sqrt{5} \sqrt{5}}{6}$

Combine using the product rule for radicals.

$\frac{33 + 3 \sqrt{55} + \sqrt{5 \cdot 11} + \sqrt{5} \sqrt{5}}{6}$

Multiply $5$ by $11$ .

$\frac{33 + 3 \sqrt{55} + \sqrt{55} + \sqrt{5} \sqrt{5}}{6}$

Combine using the product rule for radicals.

$\frac{33 + 3 \sqrt{55} + \sqrt{55} + \sqrt{5 \cdot 5}}{6}$

Multiply $5$ by $5$ .

$\frac{33 + 3 \sqrt{55} + \sqrt{55} + \sqrt{25}}{6}$

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

$\frac{33 + 3 \sqrt{55} + \sqrt{55} + \sqrt{5^{2}}}{6}$

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

$\frac{33 + 3 \sqrt{55} + \sqrt{55} + 5}{6}$

Add $33$ and $5$ .

$\frac{38 + 3 \sqrt{55} + \sqrt{55}}{6}$

Add $3 \sqrt{55}$ and $\sqrt{55}$ .

$\frac{38 + 4 \sqrt{55}}{6}$

Step 5

Cancel the common factor of $38 + 4 \sqrt{55}$ and $6$ .

Factor $2$ out of $38$ .

$\frac{2 (19) + 4 \sqrt{55}}{6}$

Factor $2$ out of $4 \sqrt{55}$ .

$\frac{2 (19) + 2 (2 \sqrt{55})}{6}$

Factor $2$ out of $2 (19) + 2 (2 \sqrt{55})$ .

$\frac{2 (19 + 2 \sqrt{55})}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$\frac{2 (19 + 2 \sqrt{55})}{2 \cdot 3}$

Cancel the common factor.

$\frac{2 (19 + 2 \sqrt{55})}{2 \cdot 3}$

Rewrite the expression.

$\frac{19 + 2 \sqrt{55}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{19 + 2 \sqrt{55}}{3}$

Decimal Form:

$11.27746565 \dots$

Evaluate (3 square root of 11+ square root of 5)/( square root of 11- square root of 5)

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