Solve sqrt{2*[(1-2/3)}*(3/14-3/14+1/2)+1:(frac{3}{4}-frac{1}{6})-frac{1}{4}]+frac{8}{21}

$$\sqrt { 2 \times [ ( 1 - \frac { 2 } { 3 } ) } \times ( \frac { 3 } { 14 } - \frac { 3 } { 14 } + \frac { 1 } { 2 } ) + 1 : ( \frac { 3 } { 4 } - \frac { 1 } { 6 } ) - \frac { 1 } { 4 } ] + \frac { 8 } { 21 }$$

Evaluate

$$\frac{\sqrt{5754}+16}{42}\approx 2.187026787$$

Short Solution Steps

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

$$\sqrt{2\left(\left(\frac{3}{3}-\frac{2}{3}\right)\left(\frac{3}{14}-\frac{3}{14}+\frac{1}{2}\right)+\frac{1}{\frac{3}{4}-\frac{1}{6}}-\frac{1}{4}\right)}+\frac{8}{21}$$

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

$$\sqrt{2\left(\frac{3-2}{3}\left(\frac{3}{14}-\frac{3}{14}+\frac{1}{2}\right)+\frac{1}{\frac{3}{4}-\frac{1}{6}}-\frac{1}{4}\right)}+\frac{8}{21}$$

Subtract 2 from 3 to get 1.

$$\sqrt{2\left(\frac{1}{3}\left(\frac{3}{14}-\frac{3}{14}+\frac{1}{2}\right)+\frac{1}{\frac{3}{4}-\frac{1}{6}}-\frac{1}{4}\right)}+\frac{8}{21}$$

Subtract \frac{3}{14} from \frac{3}{14} to get 0.

$$\sqrt{2\left(\frac{1}{3}\times \frac{1}{2}+\frac{1}{\frac{3}{4}-\frac{1}{6}}-\frac{1}{4}\right)}+\frac{8}{21}$$

Multiply \frac{1}{3} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{2\left(\frac{1\times 1}{3\times 2}+\frac{1}{\frac{3}{4}-\frac{1}{6}}-\frac{1}{4}\right)}+\frac{8}{21}$$

Do the multiplications in the fraction \frac{1\times 1}{3\times 2}.

$$\sqrt{2\left(\frac{1}{6}+\frac{1}{\frac{3}{4}-\frac{1}{6}}-\frac{1}{4}\right)}+\frac{8}{21}$$

Least common multiple of 4 and 6 is 12. Convert \frac{3}{4} and \frac{1}{6} to fractions with denominator 12.

$$\sqrt{2\left(\frac{1}{6}+\frac{1}{\frac{9}{12}-\frac{2}{12}}-\frac{1}{4}\right)}+\frac{8}{21}$$

Since \frac{9}{12} and \frac{2}{12} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{2\left(\frac{1}{6}+\frac{1}{\frac{9-2}{12}}-\frac{1}{4}\right)}+\frac{8}{21}$$

Subtract 2 from 9 to get 7.

$$\sqrt{2\left(\frac{1}{6}+\frac{1}{\frac{7}{12}}-\frac{1}{4}\right)}+\frac{8}{21}$$

Divide 1 by \frac{7}{12} by multiplying 1 by the reciprocal of \frac{7}{12}.

$$\sqrt{2\left(\frac{1}{6}+1\times \frac{12}{7}-\frac{1}{4}\right)}+\frac{8}{21}$$

Multiply 1 and \frac{12}{7} to get \frac{12}{7}.

$$\sqrt{2\left(\frac{1}{6}+\frac{12}{7}-\frac{1}{4}\right)}+\frac{8}{21}$$

Least common multiple of 6 and 7 is 42. Convert \frac{1}{6} and \frac{12}{7} to fractions with denominator 42.

$$\sqrt{2\left(\frac{7}{42}+\frac{72}{42}-\frac{1}{4}\right)}+\frac{8}{21}$$

Since \frac{7}{42} and \frac{72}{42} have the same denominator, add them by adding their numerators.

$$\sqrt{2\left(\frac{7+72}{42}-\frac{1}{4}\right)}+\frac{8}{21}$$

Add 7 and 72 to get 79.

$$\sqrt{2\left(\frac{79}{42}-\frac{1}{4}\right)}+\frac{8}{21}$$

Least common multiple of 42 and 4 is 84. Convert \frac{79}{42} and \frac{1}{4} to fractions with denominator 84.

$$\sqrt{2\left(\frac{158}{84}-\frac{21}{84}\right)}+\frac{8}{21}$$

Since \frac{158}{84} and \frac{21}{84} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{2\times \frac{158-21}{84}}+\frac{8}{21}$$

Subtract 21 from 158 to get 137.

$$\sqrt{2\times \frac{137}{84}}+\frac{8}{21}$$

Express 2\times \frac{137}{84} as a single fraction.

$$\sqrt{\frac{2\times 137}{84}}+\frac{8}{21}$$

Multiply 2 and 137 to get 274.

$$\sqrt{\frac{274}{84}}+\frac{8}{21}$$

Reduce the fraction \frac{274}{84} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{137}{42}}+\frac{8}{21}$$

Rewrite the square root of the division \sqrt{\frac{137}{42}} as the division of square roots \frac{\sqrt{137}}{\sqrt{42}}.

$$\frac{\sqrt{137}}{\sqrt{42}}+\frac{8}{21}$$

Rationalize the denominator of \frac{\sqrt{137}}{\sqrt{42}} by multiplying numerator and denominator by \sqrt{42}.

$$\frac{\sqrt{137}\sqrt{42}}{\left(\sqrt{42}\right)^{2}}+\frac{8}{21}$$

The square of \sqrt{42} is 42.

$$\frac{\sqrt{137}\sqrt{42}}{42}+\frac{8}{21}$$

To multiply \sqrt{137} and \sqrt{42}, multiply the numbers under the square root.

$$\frac{\sqrt{5754}}{42}+\frac{8}{21}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 42 and 21 is 42. Multiply \frac{8}{21} times \frac{2}{2}.

$$\frac{\sqrt{5754}}{42}+\frac{8\times 2}{42}$$

Since \frac{\sqrt{5754}}{42} and \frac{8\times 2}{42} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{5754}+8\times 2}{42}$$

Do the multiplications in \sqrt{5754}+8\times 2.

$$\frac{\sqrt{5754}+16}{42}$$

Factor

$$\frac{\sqrt{5754} + 16}{42} = 2.1870267874774174$$