Solve frac{6+sqrt{6}}{sqrt{30}+sqrt{5}}

$$\frac { 6 + \sqrt { 6 } } { \sqrt { 30 } + \sqrt { 5 } }$$

Evaluate

$$\frac{\sqrt{30}}{5}\approx 1.095445115$$

Solution Steps

Rationalize the denominator of \frac{6+\sqrt{6}}{\sqrt{30}+\sqrt{5}} by multiplying numerator and denominator by \sqrt{30}-\sqrt{5}.

$$\frac{\left(6+\sqrt{6}\right)\left(\sqrt{30}-\sqrt{5}\right)}{\left(\sqrt{30}+\sqrt{5}\right)\left(\sqrt{30}-\sqrt{5}\right)}$$

Consider \left(\sqrt{30}+\sqrt{5}\right)\left(\sqrt{30}-\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}.

$$\frac{\left(6+\sqrt{6}\right)\left(\sqrt{30}-\sqrt{5}\right)}{\left(\sqrt{30}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

Square \sqrt{30}. Square \sqrt{5}.

$$\frac{\left(6+\sqrt{6}\right)\left(\sqrt{30}-\sqrt{5}\right)}{30-5}$$

Subtract 5 from 30 to get 25.

$$\frac{\left(6+\sqrt{6}\right)\left(\sqrt{30}-\sqrt{5}\right)}{25}$$

Apply the distributive property by multiplying each term of 6+\sqrt{6} by each term of \sqrt{30}-\sqrt{5}.

$$\frac{6\sqrt{30}-6\sqrt{5}+\sqrt{6}\sqrt{30}-\sqrt{6}\sqrt{5}}{25}$$

Factor 30=6\times 5. Rewrite the square root of the product \sqrt{6\times 5} as the product of square roots \sqrt{6}\sqrt{5}.

$$\frac{6\sqrt{30}-6\sqrt{5}+\sqrt{6}\sqrt{6}\sqrt{5}-\sqrt{6}\sqrt{5}}{25}$$

Multiply \sqrt{6} and \sqrt{6} to get 6.

$$\frac{6\sqrt{30}-6\sqrt{5}+6\sqrt{5}-\sqrt{6}\sqrt{5}}{25}$$

Combine -6\sqrt{5} and 6\sqrt{5} to get 0.

$$\frac{6\sqrt{30}-\sqrt{6}\sqrt{5}}{25}$$

To multiply \sqrt{6} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{6\sqrt{30}-\sqrt{30}}{25}$$

Combine 6\sqrt{30} and -\sqrt{30} to get 5\sqrt{30}.

$$\frac{5\sqrt{30}}{25}$$

Divide 5\sqrt{30} by 25 to get \frac{1}{5}\sqrt{30}.

$$\frac{1}{5}\sqrt{30}$$