Solve frac{sqrt{3}+sqrt{5}}{sqrt{15}-1}

$$\frac{ \sqrt{ 3 } + \sqrt{ 5 } }{ \sqrt{ 15 } -1 }$$

Evaluate

$$\frac{2\sqrt{5}+3\sqrt{3}}{7}\approx 1.381184054$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}+\sqrt{5}}{\sqrt{15}-1} by multiplying numerator and denominator by \sqrt{15}+1.

$$\frac{\left(\sqrt{3}+\sqrt{5}\right)\left(\sqrt{15}+1\right)}{\left(\sqrt{15}-1\right)\left(\sqrt{15}+1\right)}$$

Consider \left(\sqrt{15}-1\right)\left(\sqrt{15}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{3}+\sqrt{5}\right)\left(\sqrt{15}+1\right)}{\left(\sqrt{15}\right)^{2}-1^{2}}$$

Square \sqrt{15}. Square 1.

$$\frac{\left(\sqrt{3}+\sqrt{5}\right)\left(\sqrt{15}+1\right)}{15-1}$$

Subtract 1 from 15 to get 14.

$$\frac{\left(\sqrt{3}+\sqrt{5}\right)\left(\sqrt{15}+1\right)}{14}$$

Apply the distributive property by multiplying each term of \sqrt{3}+\sqrt{5} by each term of \sqrt{15}+1.

$$\frac{\sqrt{3}\sqrt{15}+\sqrt{3}+\sqrt{5}\sqrt{15}+\sqrt{5}}{14}$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

$$\frac{\sqrt{3}\sqrt{3}\sqrt{5}+\sqrt{3}+\sqrt{5}\sqrt{15}+\sqrt{5}}{14}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{3\sqrt{5}+\sqrt{3}+\sqrt{5}\sqrt{15}+\sqrt{5}}{14}$$

Factor 15=5\times 3. Rewrite the square root of the product \sqrt{5\times 3} as the product of square roots \sqrt{5}\sqrt{3}.

$$\frac{3\sqrt{5}+\sqrt{3}+\sqrt{5}\sqrt{5}\sqrt{3}+\sqrt{5}}{14}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{3\sqrt{5}+\sqrt{3}+5\sqrt{3}+\sqrt{5}}{14}$$

Combine \sqrt{3} and 5\sqrt{3} to get 6\sqrt{3}.

$$\frac{3\sqrt{5}+6\sqrt{3}+\sqrt{5}}{14}$$

Combine 3\sqrt{5} and \sqrt{5} to get 4\sqrt{5}.

$$\frac{4\sqrt{5}+6\sqrt{3}}{14}$$