Solve frac{sqrt{5}}{7-sqrt{20}}

$$\frac { \sqrt { 5 } } { 7 - \sqrt { 20 } }$$

Evaluate

$$\frac{7\sqrt{5}+10}{29}\approx 0.884568132$$

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}.

$$\frac{\sqrt{5}}{7-2\sqrt{5}}$$

Rationalize the denominator of \frac{\sqrt{5}}{7-2\sqrt{5}} by multiplying numerator and denominator by 7+2\sqrt{5}.

$$\frac{\sqrt{5}\left(7+2\sqrt{5}\right)}{\left(7-2\sqrt{5}\right)\left(7+2\sqrt{5}\right)}$$

Consider \left(7-2\sqrt{5}\right)\left(7+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}.

$$\frac{\sqrt{5}\left(7+2\sqrt{5}\right)}{7^{2}-\left(-2\sqrt{5}\right)^{2}}$$

Calculate 7 to the power of 2 and get 49.

$$\frac{\sqrt{5}\left(7+2\sqrt{5}\right)}{49-\left(-2\sqrt{5}\right)^{2}}$$

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

$$\frac{\sqrt{5}\left(7+2\sqrt{5}\right)}{49-\left(-2\right)^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{\sqrt{5}\left(7+2\sqrt{5}\right)}{49-4\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{5}\left(7+2\sqrt{5}\right)}{49-4\times 5}$$

Multiply 4 and 5 to get 20.

$$\frac{\sqrt{5}\left(7+2\sqrt{5}\right)}{49-20}$$

Subtract 20 from 49 to get 29.

$$\frac{\sqrt{5}\left(7+2\sqrt{5}\right)}{29}$$

Use the distributive property to multiply \sqrt{5} by 7+2\sqrt{5}.

$$\frac{7\sqrt{5}+2\left(\sqrt{5}\right)^{2}}{29}$$

The square of \sqrt{5} is 5.

$$\frac{7\sqrt{5}+2\times 5}{29}$$

Multiply 2 and 5 to get 10.

$$\frac{7\sqrt{5}+10}{29}$$