Solve frac{sqrt{5}-3}{2-sqrt{2}}

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

Evaluate

$$\frac{\sqrt{10}}{2}-\frac{3\sqrt{2}}{2}+\sqrt{5}-3\approx -1.304113536$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{5}-3}{2-\sqrt{2}} by multiplying numerator and denominator by 2+\sqrt{2}.

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

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

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

Square 2. Square \sqrt{2}.

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

Subtract 2 from 4 to get 2.

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

Apply the distributive property by multiplying each term of \sqrt{5}-3 by each term of 2+\sqrt{2}.

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

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

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