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

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

Step 1

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

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

Step 2

Combine fractions.

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

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

Expand the denominator using the FOIL method.

$\frac{(\sqrt{3} + \sqrt{5}) (3 \sqrt{3} + \sqrt{5})}{9 {\sqrt{3}}^{2} + 3 \sqrt{15} - 3 \sqrt{15} - {\sqrt{5}}^{2}}$

Simplify.

$\frac{(\sqrt{3} + \sqrt{5}) (3 \sqrt{3} + \sqrt{5})}{22}$

Step 3

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

Apply the distributive property.

$\frac{\sqrt{3} (3 \sqrt{3} + \sqrt{5}) + \sqrt{5} (3 \sqrt{3} + \sqrt{5})}{22}$

Apply the distributive property.

$\frac{\sqrt{3} (3 \sqrt{3}) + \sqrt{3} \sqrt{5} + \sqrt{5} (3 \sqrt{3} + \sqrt{5})}{22}$

Apply the distributive property.

$\frac{\sqrt{3} (3 \sqrt{3}) + \sqrt{3} \sqrt{5} + \sqrt{5} (3 \sqrt{3}) + \sqrt{5} \sqrt{5}}{22}$

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{3} (3 \sqrt{3})$ .

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{3 ({\sqrt{3}}^{1} \sqrt{3}) + \sqrt{3} \sqrt{5} + \sqrt{5} (3 \sqrt{3}) + \sqrt{5} \sqrt{5}}{22}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{3 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) + \sqrt{3} \sqrt{5} + \sqrt{5} (3 \sqrt{3}) + \sqrt{5} \sqrt{5}}{22}$

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

$\frac{3 {\sqrt{3}}^{1 + 1} + \sqrt{3} \sqrt{5} + \sqrt{5} (3 \sqrt{3}) + \sqrt{5} \sqrt{5}}{22}$

Add $1$ and $1$ .

$\frac{3 {\sqrt{3}}^{2} + \sqrt{3} \sqrt{5} + \sqrt{5} (3 \sqrt{3}) + \sqrt{5} \sqrt{5}}{22}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{3 {(3^{\frac{1}{2}})}^{2} + \sqrt{3} \sqrt{5} + \sqrt{5} (3 \sqrt{3}) + \sqrt{5} \sqrt{5}}{22}$

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

$\frac{3 \cdot 3^{\frac{1}{2} \cdot 2} + \sqrt{3} \sqrt{5} + \sqrt{5} (3 \sqrt{3}) + \sqrt{5} \sqrt{5}}{22}$

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

$\frac{3 \cdot 3^{\frac{2}{2}} + \sqrt{3} \sqrt{5} + \sqrt{5} (3 \sqrt{3}) + \sqrt{5} \sqrt{5}}{22}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 \cdot 3^{\frac{2}{2}} + \sqrt{3} \sqrt{5} + \sqrt{5} (3 \sqrt{3}) + \sqrt{5} \sqrt{5}}{22}$

Rewrite the expression.

$\frac{3 \cdot 3^{1} + \sqrt{3} \sqrt{5} + \sqrt{5} (3 \sqrt{3}) + \sqrt{5} \sqrt{5}}{22}$

Evaluate the exponent.

$\frac{3 \cdot 3 + \sqrt{3} \sqrt{5} + \sqrt{5} (3 \sqrt{3}) + \sqrt{5} \sqrt{5}}{22}$

Multiply $3$ by $3$ .

$\frac{9 + \sqrt{3} \sqrt{5} + \sqrt{5} (3 \sqrt{3}) + \sqrt{5} \sqrt{5}}{22}$

Combine using the product rule for radicals.

$\frac{9 + \sqrt{3 \cdot 5} + \sqrt{5} (3 \sqrt{3}) + \sqrt{5} \sqrt{5}}{22}$

Multiply $3$ by $5$ .

$\frac{9 + \sqrt{15} + \sqrt{5} (3 \sqrt{3}) + \sqrt{5} \sqrt{5}}{22}$

Multiply $\sqrt{5} (3 \sqrt{3})$ .

Combine using the product rule for radicals.

$\frac{9 + \sqrt{15} + 3 \sqrt{5 \cdot 3} + \sqrt{5} \sqrt{5}}{22}$

Multiply $5$ by $3$ .

$\frac{9 + \sqrt{15} + 3 \sqrt{15} + \sqrt{5} \sqrt{5}}{22}$

Combine using the product rule for radicals.

$\frac{9 + \sqrt{15} + 3 \sqrt{15} + \sqrt{5 \cdot 5}}{22}$

Multiply $5$ by $5$ .

$\frac{9 + \sqrt{15} + 3 \sqrt{15} + \sqrt{25}}{22}$

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

$\frac{9 + \sqrt{15} + 3 \sqrt{15} + \sqrt{5^{2}}}{22}$

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

$\frac{9 + \sqrt{15} + 3 \sqrt{15} + 5}{22}$

Add $9$ and $5$ .

$\frac{14 + \sqrt{15} + 3 \sqrt{15}}{22}$

Add $\sqrt{15}$ and $3 \sqrt{15}$ .

$\frac{14 + 4 \sqrt{15}}{22}$

Step 5

Cancel the common factor of $14 + 4 \sqrt{15}$ and $22$ .

Factor $2$ out of $14$ .

$\frac{2 (7) + 4 \sqrt{15}}{22}$

Factor $2$ out of $4 \sqrt{15}$ .

$\frac{2 (7) + 2 (2 \sqrt{15})}{22}$

Factor $2$ out of $2 (7) + 2 (2 \sqrt{15})$ .

$\frac{2 (7 + 2 \sqrt{15})}{22}$

Cancel the common factors.

Factor $2$ out of $22$ .

$\frac{2 (7 + 2 \sqrt{15})}{2 \cdot 11}$

Cancel the common factor.

$\frac{2 (7 + 2 \sqrt{15})}{2 \cdot 11}$

Rewrite the expression.

$\frac{7 + 2 \sqrt{15}}{11}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{7 + 2 \sqrt{15}}{11}$

Decimal Form:

$1.34054242 \dots$

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

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