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

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

Evaluate

$$\frac{5\sqrt{2}+5\sqrt{5}-\sqrt{10}-25}{20}\approx -0.495543498$$

Solution Steps

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

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

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

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

Square \sqrt{5}. Square 5.

$$\frac{\left(\sqrt{2}-5\right)\left(\sqrt{5}-5\right)}{5-25}$$

Subtract 25 from 5 to get -20.

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

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

$$\frac{\sqrt{2}\sqrt{5}-5\sqrt{2}-5\sqrt{5}+25}{-20}$$

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

$$\frac{\sqrt{10}-5\sqrt{2}-5\sqrt{5}+25}{-20}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{10}+5\sqrt{2}+5\sqrt{5}-25}{20}$$