Solve frac{4-sqrt{8}}{1-sqrt{5}}

$$\frac { 4 - \sqrt { 8 } } { 1 - \sqrt { 5 } }$$

Evaluate

$$\frac{\sqrt{2}+\sqrt{10}-2\sqrt{5}-2}{2}\approx -0.947822366$$

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}.

$$\frac{4-2\sqrt{2}}{1-\sqrt{5}}$$

Rationalize the denominator of \frac{4-2\sqrt{2}}{1-\sqrt{5}} by multiplying numerator and denominator by 1+\sqrt{5}.

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

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}.

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

Square 1. Square \sqrt{5}.

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

Subtract 5 from 1 to get -4.

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

Apply the distributive property by multiplying each term of 4-2\sqrt{2} by each term of 1+\sqrt{5}.

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

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

$$\frac{4+4\sqrt{5}-2\sqrt{2}-2\sqrt{10}}{-4}$$