Solve sqrt{1/7}:(1/3)+(3/2):(1-1/8)-frac{3}{7}:(1-frac{1}{3})

$$\sqrt { \frac { 1 } { 7 } } : ( \frac { 1 } { 3 } ) + ( \frac { 3 } { 2 } ) : ( 1 - \frac { 1 } { 8 } ) - \frac { 3 } { 7 } : ( 1 - \frac { 1 } { 3 } )$$

Evaluate

$$\frac{3\sqrt{7}}{7}+\frac{15}{14}\approx 2.20532199$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{7}} as the division of square roots \frac{\sqrt{1}}{\sqrt{7}}.

$$\frac{\frac{\sqrt{1}}{\sqrt{7}}}{\frac{1}{3}}+\frac{\frac{3}{2}}{1-\frac{1}{8}}-\frac{\frac{3}{7}}{1-\frac{1}{3}}$$

Calculate the square root of 1 and get 1.

$$\frac{\frac{1}{\sqrt{7}}}{\frac{1}{3}}+\frac{\frac{3}{2}}{1-\frac{1}{8}}-\frac{\frac{3}{7}}{1-\frac{1}{3}}$$

Rationalize the denominator of \frac{1}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\frac{\sqrt{7}}{\left(\sqrt{7}\right)^{2}}}{\frac{1}{3}}+\frac{\frac{3}{2}}{1-\frac{1}{8}}-\frac{\frac{3}{7}}{1-\frac{1}{3}}$$

The square of \sqrt{7} is 7.

$$\frac{\frac{\sqrt{7}}{7}}{\frac{1}{3}}+\frac{\frac{3}{2}}{1-\frac{1}{8}}-\frac{\frac{3}{7}}{1-\frac{1}{3}}$$

Divide \frac{\sqrt{7}}{7} by \frac{1}{3} by multiplying \frac{\sqrt{7}}{7} by the reciprocal of \frac{1}{3}.

$$\frac{\sqrt{7}\times 3}{7}+\frac{\frac{3}{2}}{1-\frac{1}{8}}-\frac{\frac{3}{7}}{1-\frac{1}{3}}$$

Convert 1 to fraction \frac{8}{8}.

$$\frac{\sqrt{7}\times 3}{7}+\frac{\frac{3}{2}}{\frac{8}{8}-\frac{1}{8}}-\frac{\frac{3}{7}}{1-\frac{1}{3}}$$

Since \frac{8}{8} and \frac{1}{8} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{7}\times 3}{7}+\frac{\frac{3}{2}}{\frac{8-1}{8}}-\frac{\frac{3}{7}}{1-\frac{1}{3}}$$

Subtract 1 from 8 to get 7.

$$\frac{\sqrt{7}\times 3}{7}+\frac{\frac{3}{2}}{\frac{7}{8}}-\frac{\frac{3}{7}}{1-\frac{1}{3}}$$

Divide \frac{3}{2} by \frac{7}{8} by multiplying \frac{3}{2} by the reciprocal of \frac{7}{8}.

$$\frac{\sqrt{7}\times 3}{7}+\frac{3}{2}\times \frac{8}{7}-\frac{\frac{3}{7}}{1-\frac{1}{3}}$$

Multiply \frac{3}{2} times \frac{8}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{7}\times 3}{7}+\frac{3\times 8}{2\times 7}-\frac{\frac{3}{7}}{1-\frac{1}{3}}$$

Do the multiplications in the fraction \frac{3\times 8}{2\times 7}.

$$\frac{\sqrt{7}\times 3}{7}+\frac{24}{14}-\frac{\frac{3}{7}}{1-\frac{1}{3}}$$

Reduce the fraction \frac{24}{14} to lowest terms by extracting and canceling out 2.

$$\frac{\sqrt{7}\times 3}{7}+\frac{12}{7}-\frac{\frac{3}{7}}{1-\frac{1}{3}}$$

Convert 1 to fraction \frac{3}{3}.

$$\frac{\sqrt{7}\times 3}{7}+\frac{12}{7}-\frac{\frac{3}{7}}{\frac{3}{3}-\frac{1}{3}}$$

Since \frac{3}{3} and \frac{1}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{7}\times 3}{7}+\frac{12}{7}-\frac{\frac{3}{7}}{\frac{3-1}{3}}$$

Subtract 1 from 3 to get 2.

$$\frac{\sqrt{7}\times 3}{7}+\frac{12}{7}-\frac{\frac{3}{7}}{\frac{2}{3}}$$

Divide \frac{3}{7} by \frac{2}{3} by multiplying \frac{3}{7} by the reciprocal of \frac{2}{3}.

$$\frac{\sqrt{7}\times 3}{7}+\frac{12}{7}-\frac{3}{7}\times \frac{3}{2}$$

Multiply \frac{3}{7} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{7}\times 3}{7}+\frac{12}{7}-\frac{3\times 3}{7\times 2}$$

Do the multiplications in the fraction \frac{3\times 3}{7\times 2}.

$$\frac{\sqrt{7}\times 3}{7}+\frac{12}{7}-\frac{9}{14}$$

Least common multiple of 7 and 14 is 14. Convert \frac{12}{7} and \frac{9}{14} to fractions with denominator 14.

$$\frac{\sqrt{7}\times 3}{7}+\frac{24}{14}-\frac{9}{14}$$

Since \frac{24}{14} and \frac{9}{14} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{7}\times 3}{7}+\frac{24-9}{14}$$

Subtract 9 from 24 to get 15.

$$\frac{\sqrt{7}\times 3}{7}+\frac{15}{14}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 7 and 14 is 14. Multiply \frac{\sqrt{7}\times 3}{7} times \frac{2}{2}.

$$\frac{2\sqrt{7}\times 3}{14}+\frac{15}{14}$$

Since \frac{2\sqrt{7}\times 3}{14} and \frac{15}{14} have the same denominator, add them by adding their numerators.

$$\frac{2\sqrt{7}\times 3+15}{14}$$

Do the multiplications in 2\sqrt{7}\times 3+15.

$$\frac{6\sqrt{7}+15}{14}$$

Factor

$$\frac{3 {(2 \sqrt{7} + 5)}}{14} = 2.205321990456253$$