Solve frac{sqrt{8}}{1-sqrt{5}}

$$\frac{ \sqrt{ 8 } }{ 1- \sqrt{ 5 } }$$

Evaluate

$$-\frac{\sqrt{2}\left(\sqrt{5}+1\right)}{2}\approx -2.288245611$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{2\sqrt{2}}{1-\sqrt{5}}$$

Rationalize the denominator of \frac{2\sqrt{2}}{1-\sqrt{5}} by multiplying numerator and denominator by 1+\sqrt{5}.

$$\frac{2\sqrt{2}\left(1+\sqrt{5}\right)}{\left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right)}$$

Consider \left(1-\sqrt{5}\right)\left(1+\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}.

$$\frac{2\sqrt{2}\left(1+\sqrt{5}\right)}{1^{2}-\left(\sqrt{5}\right)^{2}}$$

Square 1. Square \sqrt{5}.

$$\frac{2\sqrt{2}\left(1+\sqrt{5}\right)}{1-5}$$

Subtract 5 from 1 to get -4.

$$\frac{2\sqrt{2}\left(1+\sqrt{5}\right)}{-4}$$

Use the distributive property to multiply 2\sqrt{2} by 1+\sqrt{5}.

$$\frac{2\sqrt{2}+2\sqrt{2}\sqrt{5}}{-4}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{2\sqrt{2}+2\sqrt{10}}{-4}$$