Solve sqrt{[(1/4+1/2-3/5):(9/10-frac{2}{3})]*frac{1}{7}}=

$$\sqrt { [ ( \frac { 1 } { 4 } + \frac { 1 } { 2 } - \frac { 3 } { 5 } ) : ( \frac { 9 } { 10 } - \frac { 2 } { 3 } ) ] \times \frac { 1 } { 7 } } =$$

Evaluate

$$\frac{3\sqrt{2}}{14}\approx 0.303045763$$

Solution Steps

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

$$\sqrt{\frac{\frac{1}{4}+\frac{2}{4}-\frac{3}{5}}{\frac{9}{10}-\frac{2}{3}}\times \frac{1}{7}}$$

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

$$\sqrt{\frac{\frac{1+2}{4}-\frac{3}{5}}{\frac{9}{10}-\frac{2}{3}}\times \frac{1}{7}}$$

Add 1 and 2 to get 3.

$$\sqrt{\frac{\frac{3}{4}-\frac{3}{5}}{\frac{9}{10}-\frac{2}{3}}\times \frac{1}{7}}$$

Least common multiple of 4 and 5 is 20. Convert \frac{3}{4} and \frac{3}{5} to fractions with denominator 20.

$$\sqrt{\frac{\frac{15}{20}-\frac{12}{20}}{\frac{9}{10}-\frac{2}{3}}\times \frac{1}{7}}$$

Since \frac{15}{20} and \frac{12}{20} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{15-12}{20}}{\frac{9}{10}-\frac{2}{3}}\times \frac{1}{7}}$$

Subtract 12 from 15 to get 3.

$$\sqrt{\frac{\frac{3}{20}}{\frac{9}{10}-\frac{2}{3}}\times \frac{1}{7}}$$

Least common multiple of 10 and 3 is 30. Convert \frac{9}{10} and \frac{2}{3} to fractions with denominator 30.

$$\sqrt{\frac{\frac{3}{20}}{\frac{27}{30}-\frac{20}{30}}\times \frac{1}{7}}$$

Since \frac{27}{30} and \frac{20}{30} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{3}{20}}{\frac{27-20}{30}}\times \frac{1}{7}}$$

Subtract 20 from 27 to get 7.

$$\sqrt{\frac{\frac{3}{20}}{\frac{7}{30}}\times \frac{1}{7}}$$

Divide \frac{3}{20} by \frac{7}{30} by multiplying \frac{3}{20} by the reciprocal of \frac{7}{30}.

$$\sqrt{\frac{3}{20}\times \frac{30}{7}\times \frac{1}{7}}$$

Multiply \frac{3}{20} times \frac{30}{7} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{3\times 30}{20\times 7}\times \frac{1}{7}}$$

Do the multiplications in the fraction \frac{3\times 30}{20\times 7}.

$$\sqrt{\frac{90}{140}\times \frac{1}{7}}$$

Reduce the fraction \frac{90}{140} to lowest terms by extracting and canceling out 10.

$$\sqrt{\frac{9}{14}\times \frac{1}{7}}$$

Multiply \frac{9}{14} times \frac{1}{7} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{9\times 1}{14\times 7}}$$

Do the multiplications in the fraction \frac{9\times 1}{14\times 7}.

$$\sqrt{\frac{9}{98}}$$

Rewrite the square root of the division \sqrt{\frac{9}{98}} as the division of square roots \frac{\sqrt{9}}{\sqrt{98}}.

$$\frac{\sqrt{9}}{\sqrt{98}}$$

Calculate the square root of 9 and get 3.

$$\frac{3}{\sqrt{98}}$$

Factor 98=7^{2}\times 2. Rewrite the square root of the product \sqrt{7^{2}\times 2} as the product of square roots \sqrt{7^{2}}\sqrt{2}. Take the square root of 7^{2}.

$$\frac{3}{7\sqrt{2}}$$

Rationalize the denominator of \frac{3}{7\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{3\sqrt{2}}{7\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{3\sqrt{2}}{7\times 2}$$

Multiply 7 and 2 to get 14.

$$\frac{3\sqrt{2}}{14}$$