Solve sqrt{10}left(1/sqrt{2}-1/sqrt{5}right)+sqrt{8}

$$\sqrt{ 10 } \left( \frac{ 1 }{ \sqrt{ 2 } } - \frac{ 1 }{ \sqrt{ 5 } } \right) + \sqrt{ 8 }$$

Evaluate

$$\sqrt{2}+\sqrt{5}\approx 3.65028154$$

Short Solution Steps

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

$$\sqrt{10}\left(\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\frac{1}{\sqrt{5}}\right)+\sqrt{8}$$

The square of \sqrt{2} is 2.

$$\sqrt{10}\left(\frac{\sqrt{2}}{2}-\frac{1}{\sqrt{5}}\right)+\sqrt{8}$$

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

$$\sqrt{10}\left(\frac{\sqrt{2}}{2}-\frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\right)+\sqrt{8}$$

The square of \sqrt{5} is 5.

$$\sqrt{10}\left(\frac{\sqrt{2}}{2}-\frac{\sqrt{5}}{5}\right)+\sqrt{8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 5 is 10. Multiply \frac{\sqrt{2}}{2} times \frac{5}{5}. Multiply \frac{\sqrt{5}}{5} times \frac{2}{2}.

$$\sqrt{10}\left(\frac{5\sqrt{2}}{10}-\frac{2\sqrt{5}}{10}\right)+\sqrt{8}$$

Since \frac{5\sqrt{2}}{10} and \frac{2\sqrt{5}}{10} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{10}\times \frac{5\sqrt{2}-2\sqrt{5}}{10}+\sqrt{8}$$

Express \sqrt{10}\times \frac{5\sqrt{2}-2\sqrt{5}}{10} as a single fraction.

$$\frac{\sqrt{10}\left(5\sqrt{2}-2\sqrt{5}\right)}{10}+\sqrt{8}$$

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}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{2} times \frac{10}{10}.

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

Since \frac{\sqrt{10}\left(5\sqrt{2}-2\sqrt{5}\right)}{10} and \frac{10\times 2\sqrt{2}}{10} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \sqrt{10}\left(5\sqrt{2}-2\sqrt{5}\right)+10\times 2\sqrt{2}.

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

Do the calculations in 10\sqrt{5}-10\sqrt{2}+20\sqrt{2}.

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

Divide each term of 10\sqrt{5}+10\sqrt{2} by 10 to get \sqrt{5}+\sqrt{2}.

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