Solve frac{sqrt{15}}{sqrt{5}+sqrt{3}}=

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

Evaluate

$$\frac{\sqrt{5}\left(\sqrt{15}-3\right)}{2}\approx 0.976025053$$

Solution Steps

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

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

Consider \left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{5}. Square \sqrt{3}.

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

Subtract 3 from 5 to get 2.

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

Use the distributive property to multiply \sqrt{15} by \sqrt{5}-\sqrt{3}.

$$\frac{\sqrt{15}\sqrt{5}-\sqrt{15}\sqrt{3}}{2}$$

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}.

$$\frac{\sqrt{5}\sqrt{3}\sqrt{5}-\sqrt{15}\sqrt{3}}{2}$$

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

$$\frac{5\sqrt{3}-\sqrt{15}\sqrt{3}}{2}$$

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}.

$$\frac{5\sqrt{3}-\sqrt{3}\sqrt{5}\sqrt{3}}{2}$$

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

$$\frac{5\sqrt{3}-3\sqrt{5}}{2}$$