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

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

Evaluate

$$\frac{\sqrt{5}-\sqrt{3}}{2}\approx 0.252008585$$

Solution Steps

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

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

Consider \left(\sqrt{15}+5\right)\left(\sqrt{15}-5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{15}. Square 5.

$$\frac{\sqrt{5}\left(\sqrt{15}-5\right)}{15-25}$$

Subtract 25 from 15 to get -10.

$$\frac{\sqrt{5}\left(\sqrt{15}-5\right)}{-10}$$

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

$$\frac{\sqrt{5}\sqrt{15}-5\sqrt{5}}{-10}$$

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}.

$$\frac{\sqrt{5}\sqrt{5}\sqrt{3}-5\sqrt{5}}{-10}$$

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

$$\frac{5\sqrt{3}-5\sqrt{5}}{-10}$$