Solve sqrt{50}-1/sqrt{5}+2sqrt{20}-sqrt{45}+frac{sqrt{2}}{}2

$$\sqrt{ 50 } - \frac{ 1 }{ \sqrt{ 5 } } +2 \sqrt{ 20 } - \sqrt{ 45 } + \frac{ \sqrt{ 2 } }{ } 2$$

Evaluate

$$\frac{4\sqrt{5}}{5}+7\sqrt{2}\approx 11.688349319$$

Solution Steps

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$5\sqrt{2}-\frac{1}{\sqrt{5}}+2\sqrt{20}-\sqrt{45}+2\times \frac{\sqrt{2}}{1}$$

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

$$5\sqrt{2}-\frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}+2\sqrt{20}-\sqrt{45}+2\times \frac{\sqrt{2}}{1}$$

The square of \sqrt{5} is 5.

$$5\sqrt{2}-\frac{\sqrt{5}}{5}+2\sqrt{20}-\sqrt{45}+2\times \frac{\sqrt{2}}{1}$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$5\sqrt{2}-\frac{\sqrt{5}}{5}+2\times 2\sqrt{5}-\sqrt{45}+2\times \frac{\sqrt{2}}{1}$$

Multiply 2 and 2 to get 4.

$$5\sqrt{2}-\frac{\sqrt{5}}{5}+4\sqrt{5}-\sqrt{45}+2\times \frac{\sqrt{2}}{1}$$

Combine -\frac{\sqrt{5}}{5} and 4\sqrt{5} to get \frac{19}{5}\sqrt{5}.

$$5\sqrt{2}+\frac{19}{5}\sqrt{5}-\sqrt{45}+2\times \frac{\sqrt{2}}{1}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$5\sqrt{2}+\frac{19}{5}\sqrt{5}-3\sqrt{5}+2\times \frac{\sqrt{2}}{1}$$

Combine \frac{19}{5}\sqrt{5} and -3\sqrt{5} to get \frac{4}{5}\sqrt{5}.

$$5\sqrt{2}+\frac{4}{5}\sqrt{5}+2\times \frac{\sqrt{2}}{1}$$

Anything divided by one gives itself.

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

Combine 5\sqrt{2} and 2\sqrt{2} to get 7\sqrt{2}.

$$7\sqrt{2}+\frac{4}{5}\sqrt{5}$$