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

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

Evaluate

$$2-\sqrt{3}\approx 0.267949192$$

Solution Steps

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

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

Consider \left(\sqrt{15}+\sqrt{5}\right)\left(\sqrt{15}-\sqrt{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}.

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

Square \sqrt{15}. Square \sqrt{5}.

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

Subtract 5 from 15 to get 10.

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

Multiply \sqrt{15}-\sqrt{5} and \sqrt{15}-\sqrt{5} to get \left(\sqrt{15}-\sqrt{5}\right)^{2}.

$$\frac{\left(\sqrt{15}-\sqrt{5}\right)^{2}}{10}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{15}-\sqrt{5}\right)^{2}.

$$\frac{\left(\sqrt{15}\right)^{2}-2\sqrt{15}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{10}$$

The square of \sqrt{15} is 15.

$$\frac{15-2\sqrt{15}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{10}$$

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}.

$$\frac{15-2\sqrt{5}\sqrt{3}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{10}$$

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

$$\frac{15-2\times 5\sqrt{3}+\left(\sqrt{5}\right)^{2}}{10}$$

Multiply -2 and 5 to get -10.

$$\frac{15-10\sqrt{3}+\left(\sqrt{5}\right)^{2}}{10}$$

The square of \sqrt{5} is 5.

$$\frac{15-10\sqrt{3}+5}{10}$$

Add 15 and 5 to get 20.

$$\frac{20-10\sqrt{3}}{10}$$

Divide each term of 20-10\sqrt{3} by 10 to get 2-\sqrt{3}.

$$2-\sqrt{3}$$