Solve 15/sqrt{10+sqrt{20}}

$$\frac { 15 } { \sqrt { 10 } + \sqrt { 20 } }$$

Evaluate

$$-\frac{3\sqrt{10}}{2}+3\sqrt{5}\approx 1.964787442$$

Solution Steps

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{15}{\sqrt{10}+2\sqrt{5}}$$

Rationalize the denominator of \frac{15}{\sqrt{10}+2\sqrt{5}} by multiplying numerator and denominator by \sqrt{10}-2\sqrt{5}.

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

Consider \left(\sqrt{10}+2\sqrt{5}\right)\left(\sqrt{10}-2\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}.

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

The square of \sqrt{10} is 10.

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

Expand \left(2\sqrt{5}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{5} is 5.

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

Multiply 4 and 5 to get 20.

$$\frac{15\left(\sqrt{10}-2\sqrt{5}\right)}{10-20}$$

Subtract 20 from 10 to get -10.

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

Divide 15\left(\sqrt{10}-2\sqrt{5}\right) by -10 to get -\frac{3}{2}\left(\sqrt{10}-2\sqrt{5}\right).

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

Use the distributive property to multiply -\frac{3}{2} by \sqrt{10}-2\sqrt{5}.

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

Express -\frac{3}{2}\left(-2\right) as a single fraction.

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

Multiply -3 and -2 to get 6.

$$-\frac{3}{2}\sqrt{10}+\frac{6}{2}\sqrt{5}$$

Divide 6 by 2 to get 3.

$$-\frac{3}{2}\sqrt{10}+3\sqrt{5}$$