Solve frac{-sqrt{240}}{sqrt{5}-3sqrt{10}}

$$\frac { - \sqrt { 240 } } { \sqrt { 5 } - 3 \sqrt { 10 } }$$

Evaluate

$$\frac{4\sqrt{3}\left(3\sqrt{2}+1\right)}{17}\approx 2.13659295$$

Solution Steps

Factor 240=4^{2}\times 15. Rewrite the square root of the product \sqrt{4^{2}\times 15} as the product of square roots \sqrt{4^{2}}\sqrt{15}. Take the square root of 4^{2}.

$$\frac{-4\sqrt{15}}{\sqrt{5}-3\sqrt{10}}$$

Rationalize the denominator of \frac{-4\sqrt{15}}{\sqrt{5}-3\sqrt{10}} by multiplying numerator and denominator by \sqrt{5}+3\sqrt{10}.

$$\frac{\left(-4\sqrt{15}\right)\left(\sqrt{5}+3\sqrt{10}\right)}{\left(\sqrt{5}-3\sqrt{10}\right)\left(\sqrt{5}+3\sqrt{10}\right)}$$

Consider \left(\sqrt{5}-3\sqrt{10}\right)\left(\sqrt{5}+3\sqrt{10}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(-4\sqrt{15}\right)\left(\sqrt{5}+3\sqrt{10}\right)}{\left(\sqrt{5}\right)^{2}-\left(-3\sqrt{10}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(-4\sqrt{15}\right)\left(\sqrt{5}+3\sqrt{10}\right)}{5-\left(-3\sqrt{10}\right)^{2}}$$

Expand \left(-3\sqrt{10}\right)^{2}.

$$\frac{\left(-4\sqrt{15}\right)\left(\sqrt{5}+3\sqrt{10}\right)}{5-\left(-3\right)^{2}\left(\sqrt{10}\right)^{2}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{\left(-4\sqrt{15}\right)\left(\sqrt{5}+3\sqrt{10}\right)}{5-9\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\left(-4\sqrt{15}\right)\left(\sqrt{5}+3\sqrt{10}\right)}{5-9\times 10}$$

Multiply 9 and 10 to get 90.

$$\frac{\left(-4\sqrt{15}\right)\left(\sqrt{5}+3\sqrt{10}\right)}{5-90}$$

Subtract 90 from 5 to get -85.

$$\frac{\left(-4\sqrt{15}\right)\left(\sqrt{5}+3\sqrt{10}\right)}{-85}$$

Use the distributive property to multiply -4\sqrt{15} by \sqrt{5}+3\sqrt{10}.

$$\frac{\left(-4\sqrt{15}\right)\sqrt{5}+3\left(-4\sqrt{15}\right)\sqrt{10}}{-85}$$

Multiply -1 and 4 to get -4.

$$\frac{-4\sqrt{15}\sqrt{5}+3\left(-1\right)\times 4\sqrt{15}\sqrt{10}}{-85}$$

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}.

$$\frac{-4\sqrt{5}\sqrt{3}\sqrt{5}+3\left(-1\right)\times 4\sqrt{15}\sqrt{10}}{-85}$$

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

$$\frac{-4\times 5\sqrt{3}+3\left(-1\right)\times 4\sqrt{15}\sqrt{10}}{-85}$$

Multiply 3 and -1 to get -3.

$$\frac{-4\times 5\sqrt{3}-3\times 4\sqrt{15}\sqrt{10}}{-85}$$

Multiply -3 and 4 to get -12.

$$\frac{-4\times 5\sqrt{3}-12\sqrt{15}\sqrt{10}}{-85}$$

To multiply \sqrt{15} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{-4\times 5\sqrt{3}-12\sqrt{150}}{-85}$$

Multiply -4 and 5 to get -20.

$$\frac{-20\sqrt{3}-12\sqrt{150}}{-85}$$

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$\frac{-20\sqrt{3}-12\times 5\sqrt{6}}{-85}$$

Multiply -12 and 5 to get -60.

$$\frac{-20\sqrt{3}-60\sqrt{6}}{-85}$$