Solve sqrt{[(3/4-7/4*1/4)*8/5+(1+frac{3}{8}:frac{1}{4})*frac{3}{10}]*frac{1}{4}+frac{5}{4}}

$$\sqrt { [ ( \frac { 3 } { 4 } - \frac { 7 } { 4 } \times \frac { 1 } { 4 } ) \times \frac { 8 } { 5 } + ( 1 + \frac { 3 } { 8 } : \frac { 1 } { 4 } ) \times \frac { 3 } { 10 } ] \times \frac { 1 } { 4 } + \frac { 5 } { 4 } }$$

Evaluate

$$\frac{5}{4}=1.25$$

Solution Steps

Multiply \frac{7}{4} times \frac{1}{4} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\left(\left(\frac{3}{4}-\frac{7\times 1}{4\times 4}\right)\times \frac{8}{5}+\left(1+\frac{\frac{3}{8}}{\frac{1}{4}}\right)\times \frac{3}{10}\right)\times \frac{1}{4}+\frac{5}{4}}$$

Do the multiplications in the fraction \frac{7\times 1}{4\times 4}.

$$\sqrt{\left(\left(\frac{3}{4}-\frac{7}{16}\right)\times \frac{8}{5}+\left(1+\frac{\frac{3}{8}}{\frac{1}{4}}\right)\times \frac{3}{10}\right)\times \frac{1}{4}+\frac{5}{4}}$$

Least common multiple of 4 and 16 is 16. Convert \frac{3}{4} and \frac{7}{16} to fractions with denominator 16.

$$\sqrt{\left(\left(\frac{12}{16}-\frac{7}{16}\right)\times \frac{8}{5}+\left(1+\frac{\frac{3}{8}}{\frac{1}{4}}\right)\times \frac{3}{10}\right)\times \frac{1}{4}+\frac{5}{4}}$$

Since \frac{12}{16} and \frac{7}{16} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\left(\frac{12-7}{16}\times \frac{8}{5}+\left(1+\frac{\frac{3}{8}}{\frac{1}{4}}\right)\times \frac{3}{10}\right)\times \frac{1}{4}+\frac{5}{4}}$$

Subtract 7 from 12 to get 5.

$$\sqrt{\left(\frac{5}{16}\times \frac{8}{5}+\left(1+\frac{\frac{3}{8}}{\frac{1}{4}}\right)\times \frac{3}{10}\right)\times \frac{1}{4}+\frac{5}{4}}$$

Multiply \frac{5}{16} times \frac{8}{5} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\left(\frac{5\times 8}{16\times 5}+\left(1+\frac{\frac{3}{8}}{\frac{1}{4}}\right)\times \frac{3}{10}\right)\times \frac{1}{4}+\frac{5}{4}}$$

Cancel out 5 in both numerator and denominator.

$$\sqrt{\left(\frac{8}{16}+\left(1+\frac{\frac{3}{8}}{\frac{1}{4}}\right)\times \frac{3}{10}\right)\times \frac{1}{4}+\frac{5}{4}}$$

Reduce the fraction \frac{8}{16} to lowest terms by extracting and canceling out 8.

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

Divide \frac{3}{8} by \frac{1}{4} by multiplying \frac{3}{8} by the reciprocal of \frac{1}{4}.

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

Express \frac{3}{8}\times 4 as a single fraction.

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

Multiply 3 and 4 to get 12.

$$\sqrt{\left(\frac{1}{2}+\left(1+\frac{12}{8}\right)\times \frac{3}{10}\right)\times \frac{1}{4}+\frac{5}{4}}$$

Reduce the fraction \frac{12}{8} to lowest terms by extracting and canceling out 4.

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

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

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

Since \frac{2}{2} and \frac{3}{2} have the same denominator, add them by adding their numerators.

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

Add 2 and 3 to get 5.

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

Multiply \frac{5}{2} times \frac{3}{10} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{5\times 3}{2\times 10}.

$$\sqrt{\left(\frac{1}{2}+\frac{15}{20}\right)\times \frac{1}{4}+\frac{5}{4}}$$

Reduce the fraction \frac{15}{20} to lowest terms by extracting and canceling out 5.

$$\sqrt{\left(\frac{1}{2}+\frac{3}{4}\right)\times \frac{1}{4}+\frac{5}{4}}$$

Least common multiple of 2 and 4 is 4. Convert \frac{1}{2} and \frac{3}{4} to fractions with denominator 4.

$$\sqrt{\left(\frac{2}{4}+\frac{3}{4}\right)\times \frac{1}{4}+\frac{5}{4}}$$

Since \frac{2}{4} and \frac{3}{4} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{2+3}{4}\times \frac{1}{4}+\frac{5}{4}}$$

Add 2 and 3 to get 5.

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

Multiply \frac{5}{4} times \frac{1}{4} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{5\times 1}{4\times 4}.

$$\sqrt{\frac{5}{16}+\frac{5}{4}}$$

Least common multiple of 16 and 4 is 16. Convert \frac{5}{16} and \frac{5}{4} to fractions with denominator 16.

$$\sqrt{\frac{5}{16}+\frac{20}{16}}$$

Since \frac{5}{16} and \frac{20}{16} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{5+20}{16}}$$

Add 5 and 20 to get 25.

$$\sqrt{\frac{25}{16}}$$

Rewrite the square root of the division \frac{25}{16} as the division of square roots \frac{\sqrt{25}}{\sqrt{16}}. Take the square root of both numerator and denominator.

$$\frac{5}{4}$$

Factor

$$\frac{5}{2 ^ {2}} = 1\frac{1}{4} = 1.25$$