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

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

Evaluate

$$\frac{5\sqrt{10}}{8}\approx 1.976423538$$

Solution Steps

Least common multiple of 8 and 4 is 8. Convert \frac{3}{8} and \frac{5}{4} to fractions with denominator 8.

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

Since \frac{3}{8} and \frac{10}{8} have the same denominator, add them by adding their numerators.

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

Add 3 and 10 to get 13.

$$\sqrt{\left(\frac{13}{8}+\frac{3}{2}\right)\left(2+\frac{1}{2}\right)\left(1-\frac{1}{2}\right)}$$

Least common multiple of 8 and 2 is 8. Convert \frac{13}{8} and \frac{3}{2} to fractions with denominator 8.

$$\sqrt{\left(\frac{13}{8}+\frac{12}{8}\right)\left(2+\frac{1}{2}\right)\left(1-\frac{1}{2}\right)}$$

Since \frac{13}{8} and \frac{12}{8} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{13+12}{8}\left(2+\frac{1}{2}\right)\left(1-\frac{1}{2}\right)}$$

Add 13 and 12 to get 25.

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

Convert 2 to fraction \frac{4}{2}.

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

Since \frac{4}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

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

Add 4 and 1 to get 5.

$$\sqrt{\frac{25}{8}\times \frac{5}{2}\left(1-\frac{1}{2}\right)}$$

Multiply \frac{25}{8} times \frac{5}{2} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{25\times 5}{8\times 2}\left(1-\frac{1}{2}\right)}$$

Do the multiplications in the fraction \frac{25\times 5}{8\times 2}.

$$\sqrt{\frac{125}{16}\left(1-\frac{1}{2}\right)}$$

Convert 1 to fraction \frac{2}{2}.

$$\sqrt{\frac{125}{16}\left(\frac{2}{2}-\frac{1}{2}\right)}$$

Since \frac{2}{2} and \frac{1}{2} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{125}{16}\times \frac{2-1}{2}}$$

Subtract 1 from 2 to get 1.

$$\sqrt{\frac{125}{16}\times \frac{1}{2}}$$

Multiply \frac{125}{16} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{125\times 1}{16\times 2}}$$

Do the multiplications in the fraction \frac{125\times 1}{16\times 2}.

$$\sqrt{\frac{125}{32}}$$

Rewrite the square root of the division \sqrt{\frac{125}{32}} as the division of square roots \frac{\sqrt{125}}{\sqrt{32}}.

$$\frac{\sqrt{125}}{\sqrt{32}}$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

$$\frac{5\sqrt{5}}{\sqrt{32}}$$

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

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

Rationalize the denominator of \frac{5\sqrt{5}}{4\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{5\sqrt{5}\sqrt{2}}{4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

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

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

$$\frac{5\sqrt{10}}{4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{5\sqrt{10}}{8}$$