Solve 5sqrt{1/5}-5/4sqrt{4/5}+sqrt{45}

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

Evaluate

$$\frac{7\sqrt{5}}{2}\approx 7.826237921$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{5}} as the division of square roots \frac{\sqrt{1}}{\sqrt{5}}.

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

Calculate the square root of 1 and get 1.

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

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

$$5\times \frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}-\frac{5}{4}\sqrt{\frac{4}{5}}+\sqrt{45}$$

The square of \sqrt{5} is 5.

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

Cancel out 5 and 5.

$$\sqrt{5}-\frac{5}{4}\sqrt{\frac{4}{5}}+\sqrt{45}$$

Rewrite the square root of the division \sqrt{\frac{4}{5}} as the division of square roots \frac{\sqrt{4}}{\sqrt{5}}.

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

Calculate the square root of 4 and get 2.

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

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

$$\sqrt{5}-\frac{5}{4}\times \frac{2\sqrt{5}}{\left(\sqrt{5}\right)^{2}}+\sqrt{45}$$

The square of \sqrt{5} is 5.

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

Multiply -\frac{5}{4} times \frac{2\sqrt{5}}{5} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2\times 5 in both numerator and denominator.

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

Combine \sqrt{5} and \frac{-\sqrt{5}}{2} to get \frac{1}{2}\sqrt{5}.

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

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}.

$$\frac{1}{2}\sqrt{5}+3\sqrt{5}$$

Combine \frac{1}{2}\sqrt{5} and 3\sqrt{5} to get \frac{7}{2}\sqrt{5}.

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