Solve sqrt{(2+1/2)^2*(1/2+1-3/5)-(3/8+frac{1}{4}-frac{3}{5}+frac{1}{10})+(frac{3}{8}+frac{1}{4})}

$$\sqrt { ( 2 + \frac { 1 } { 2 } ) ^ { 2 } \times ( \frac { 1 } { 2 } + 1 - \frac { 3 } { 5 } ) - ( \frac { 3 } { 8 } + \frac { 1 } { 4 } - \frac { 3 } { 5 } + \frac { 1 } { 10 } ) + ( \frac { 3 } { 8 } + \frac { 1 } { 4 } ) }$$

Evaluate

$$\frac{7\sqrt{2}}{4}\approx 2.474873734$$

Solution Steps

Add 2 and \frac{1}{2} to get \frac{5}{2}.

$$\sqrt{\left(\frac{5}{2}\right)^{2}\left(\frac{1}{2}+1-\frac{3}{5}\right)-\left(\frac{3}{8}+\frac{1}{4}-\frac{3}{5}+\frac{1}{10}\right)+\frac{3}{8}+\frac{1}{4}}$$

Calculate \frac{5}{2} to the power of 2 and get \frac{25}{4}.

$$\sqrt{\frac{25}{4}\left(\frac{1}{2}+1-\frac{3}{5}\right)-\left(\frac{3}{8}+\frac{1}{4}-\frac{3}{5}+\frac{1}{10}\right)+\frac{3}{8}+\frac{1}{4}}$$

Add \frac{1}{2} and 1 to get \frac{3}{2}.

$$\sqrt{\frac{25}{4}\left(\frac{3}{2}-\frac{3}{5}\right)-\left(\frac{3}{8}+\frac{1}{4}-\frac{3}{5}+\frac{1}{10}\right)+\frac{3}{8}+\frac{1}{4}}$$

Subtract \frac{3}{5} from \frac{3}{2} to get \frac{9}{10}.

$$\sqrt{\frac{25}{4}\times \frac{9}{10}-\left(\frac{3}{8}+\frac{1}{4}-\frac{3}{5}+\frac{1}{10}\right)+\frac{3}{8}+\frac{1}{4}}$$

Multiply \frac{25}{4} and \frac{9}{10} to get \frac{45}{8}.

$$\sqrt{\frac{45}{8}-\left(\frac{3}{8}+\frac{1}{4}-\frac{3}{5}+\frac{1}{10}\right)+\frac{3}{8}+\frac{1}{4}}$$

Add \frac{3}{8} and \frac{1}{4} to get \frac{5}{8}.

$$\sqrt{\frac{45}{8}-\left(\frac{5}{8}-\frac{3}{5}+\frac{1}{10}\right)+\frac{3}{8}+\frac{1}{4}}$$

Subtract \frac{3}{5} from \frac{5}{8} to get \frac{1}{40}.

$$\sqrt{\frac{45}{8}-\left(\frac{1}{40}+\frac{1}{10}\right)+\frac{3}{8}+\frac{1}{4}}$$

Add \frac{1}{40} and \frac{1}{10} to get \frac{1}{8}.

$$\sqrt{\frac{45}{8}-\frac{1}{8}+\frac{3}{8}+\frac{1}{4}}$$

Subtract \frac{1}{8} from \frac{45}{8} to get \frac{11}{2}.

$$\sqrt{\frac{11}{2}+\frac{3}{8}+\frac{1}{4}}$$

Add \frac{11}{2} and \frac{3}{8} to get \frac{47}{8}.

$$\sqrt{\frac{47}{8}+\frac{1}{4}}$$

Add \frac{47}{8} and \frac{1}{4} to get \frac{49}{8}.

$$\sqrt{\frac{49}{8}}$$

Rewrite the square root of the division \sqrt{\frac{49}{8}} as the division of square roots \frac{\sqrt{49}}{\sqrt{8}}.

$$\frac{\sqrt{49}}{\sqrt{8}}$$

Calculate the square root of 49 and get 7.

$$\frac{7}{\sqrt{8}}$$

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}.

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

Rationalize the denominator of \frac{7}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{7\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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