Solve frac{sqrt{30}}{sqrt{5}+3sqrt{2}}

$$\frac { \sqrt { 30 } } { \sqrt { 5 } + 3 \sqrt { 2 } }$$

Evaluate

$$\frac{\sqrt{6}\left(3\sqrt{10}-5\right)}{13}\approx 0.845419336$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{30}}{\sqrt{5}+3\sqrt{2}} by multiplying numerator and denominator by \sqrt{5}-3\sqrt{2}.

$$\frac{\sqrt{30}\left(\sqrt{5}-3\sqrt{2}\right)}{\left(\sqrt{5}+3\sqrt{2}\right)\left(\sqrt{5}-3\sqrt{2}\right)}$$

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

$$\frac{\sqrt{30}\left(\sqrt{5}-3\sqrt{2}\right)}{\left(\sqrt{5}\right)^{2}-\left(3\sqrt{2}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{30}\left(\sqrt{5}-3\sqrt{2}\right)}{5-\left(3\sqrt{2}\right)^{2}}$$

Expand \left(3\sqrt{2}\right)^{2}.

$$\frac{\sqrt{30}\left(\sqrt{5}-3\sqrt{2}\right)}{5-3^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\sqrt{30}\left(\sqrt{5}-3\sqrt{2}\right)}{5-9\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{30}\left(\sqrt{5}-3\sqrt{2}\right)}{5-9\times 2}$$

Multiply 9 and 2 to get 18.

$$\frac{\sqrt{30}\left(\sqrt{5}-3\sqrt{2}\right)}{5-18}$$

Subtract 18 from 5 to get -13.

$$\frac{\sqrt{30}\left(\sqrt{5}-3\sqrt{2}\right)}{-13}$$

Use the distributive property to multiply \sqrt{30} by \sqrt{5}-3\sqrt{2}.

$$\frac{\sqrt{30}\sqrt{5}-3\sqrt{30}\sqrt{2}}{-13}$$

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

$$\frac{\sqrt{5}\sqrt{6}\sqrt{5}-3\sqrt{30}\sqrt{2}}{-13}$$

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

$$\frac{5\sqrt{6}-3\sqrt{30}\sqrt{2}}{-13}$$

Factor 30=2\times 15. Rewrite the square root of the product \sqrt{2\times 15} as the product of square roots \sqrt{2}\sqrt{15}.

$$\frac{5\sqrt{6}-3\sqrt{2}\sqrt{15}\sqrt{2}}{-13}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{5\sqrt{6}-3\times 2\sqrt{15}}{-13}$$

Multiply -3 and 2 to get -6.

$$\frac{5\sqrt{6}-6\sqrt{15}}{-13}$$

Multiply both numerator and denominator by -1.

$$\frac{-5\sqrt{6}+6\sqrt{15}}{13}$$