Solve frac{5-sqrt{5}}{sqrt{10}-5sqrt{2}}

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

Evaluate

$$-\frac{\sqrt{2}}{2}\approx -0.707106781$$

Solution Steps

Rationalize the denominator of \frac{5-\sqrt{5}}{\sqrt{10}-5\sqrt{2}} by multiplying numerator and denominator by \sqrt{10}+5\sqrt{2}.

$$\frac{\left(5-\sqrt{5}\right)\left(\sqrt{10}+5\sqrt{2}\right)}{\left(\sqrt{10}-5\sqrt{2}\right)\left(\sqrt{10}+5\sqrt{2}\right)}$$

Consider \left(\sqrt{10}-5\sqrt{2}\right)\left(\sqrt{10}+5\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}.

$$\frac{\left(5-\sqrt{5}\right)\left(\sqrt{10}+5\sqrt{2}\right)}{\left(\sqrt{10}\right)^{2}-\left(-5\sqrt{2}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\left(5-\sqrt{5}\right)\left(\sqrt{10}+5\sqrt{2}\right)}{10-\left(-5\sqrt{2}\right)^{2}}$$

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

$$\frac{\left(5-\sqrt{5}\right)\left(\sqrt{10}+5\sqrt{2}\right)}{10-\left(-5\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate -5 to the power of 2 and get 25.

$$\frac{\left(5-\sqrt{5}\right)\left(\sqrt{10}+5\sqrt{2}\right)}{10-25\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(5-\sqrt{5}\right)\left(\sqrt{10}+5\sqrt{2}\right)}{10-25\times 2}$$

Multiply 25 and 2 to get 50.

$$\frac{\left(5-\sqrt{5}\right)\left(\sqrt{10}+5\sqrt{2}\right)}{10-50}$$

Subtract 50 from 10 to get -40.

$$\frac{\left(5-\sqrt{5}\right)\left(\sqrt{10}+5\sqrt{2}\right)}{-40}$$

Apply the distributive property by multiplying each term of 5-\sqrt{5} by each term of \sqrt{10}+5\sqrt{2}.

$$\frac{5\sqrt{10}+25\sqrt{2}-\sqrt{5}\sqrt{10}-5\sqrt{5}\sqrt{2}}{-40}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$\frac{5\sqrt{10}+25\sqrt{2}-\sqrt{5}\sqrt{5}\sqrt{2}-5\sqrt{5}\sqrt{2}}{-40}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{5\sqrt{10}+25\sqrt{2}-5\sqrt{2}-5\sqrt{5}\sqrt{2}}{-40}$$

Combine 25\sqrt{2} and -5\sqrt{2} to get 20\sqrt{2}.

$$\frac{5\sqrt{10}+20\sqrt{2}-5\sqrt{5}\sqrt{2}}{-40}$$

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

$$\frac{5\sqrt{10}+20\sqrt{2}-5\sqrt{10}}{-40}$$

Combine 5\sqrt{10} and -5\sqrt{10} to get 0.

$$\frac{20\sqrt{2}}{-40}$$

Divide 20\sqrt{2} by -40 to get -\frac{1}{2}\sqrt{2}.

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