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

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

Evaluate

$$\frac{\sqrt{15}}{3}\approx 1.290994449$$

Solution Steps

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

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

Consider \left(\sqrt{15}-\sqrt{3}\right)\left(\sqrt{15}+\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}.

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

Square \sqrt{15}. Square \sqrt{3}.

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

Subtract 3 from 15 to get 12.

$$\frac{\left(5-\sqrt{5}\right)\left(\sqrt{15}+\sqrt{3}\right)}{12}$$

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

$$\frac{5\sqrt{15}+5\sqrt{3}-\sqrt{5}\sqrt{15}-\sqrt{5}\sqrt{3}}{12}$$

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}.

$$\frac{5\sqrt{15}+5\sqrt{3}-\sqrt{5}\sqrt{5}\sqrt{3}-\sqrt{5}\sqrt{3}}{12}$$

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

$$\frac{5\sqrt{15}+5\sqrt{3}-5\sqrt{3}-\sqrt{5}\sqrt{3}}{12}$$

Combine 5\sqrt{3} and -5\sqrt{3} to get 0.

$$\frac{5\sqrt{15}-\sqrt{5}\sqrt{3}}{12}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{5\sqrt{15}-\sqrt{15}}{12}$$

Combine 5\sqrt{15} and -\sqrt{15} to get 4\sqrt{15}.

$$\frac{4\sqrt{15}}{12}$$

Divide 4\sqrt{15} by 12 to get \frac{1}{3}\sqrt{15}.

$$\frac{1}{3}\sqrt{15}$$