Solve sqrt{{[(15/7)^2:9/49]*[(1/10+3/20+frac{1}{10})*frac{4}{5}]*frac{1}{14}}:frac{18}{32}}

$$\sqrt { \{ [ ( \frac { 15 } { 7 } ) ^ { 2 } : \frac { 9 } { 49 } ] \times [ ( \frac { 1 } { 10 } + \frac { 3 } { 20 } + \frac { 1 } { 10 } ) \times \frac { 4 } { 5 } ] \times \frac { 1 } { 14 } \} : \frac { 18 } { 32 } }$$

Evaluate

$$\frac{2\sqrt{2}}{3}\approx 0.942809042$$

Solution Steps

Calculate \frac{15}{7} to the power of 2 and get \frac{225}{49}.

$$\sqrt{\frac{\frac{\frac{225}{49}}{\frac{9}{49}}\left(\frac{1}{10}+\frac{3}{20}+\frac{1}{10}\right)\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Divide \frac{225}{49} by \frac{9}{49} by multiplying \frac{225}{49} by the reciprocal of \frac{9}{49}.

$$\sqrt{\frac{\frac{225}{49}\times \frac{49}{9}\left(\frac{1}{10}+\frac{3}{20}+\frac{1}{10}\right)\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Multiply \frac{225}{49} times \frac{49}{9} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{225\times 49}{49\times 9}\left(\frac{1}{10}+\frac{3}{20}+\frac{1}{10}\right)\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Cancel out 49 in both numerator and denominator.

$$\sqrt{\frac{\frac{225}{9}\left(\frac{1}{10}+\frac{3}{20}+\frac{1}{10}\right)\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Divide 225 by 9 to get 25.

$$\sqrt{\frac{25\left(\frac{1}{10}+\frac{3}{20}+\frac{1}{10}\right)\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Least common multiple of 10 and 20 is 20. Convert \frac{1}{10} and \frac{3}{20} to fractions with denominator 20.

$$\sqrt{\frac{25\left(\frac{2}{20}+\frac{3}{20}+\frac{1}{10}\right)\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Since \frac{2}{20} and \frac{3}{20} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{25\left(\frac{2+3}{20}+\frac{1}{10}\right)\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Add 2 and 3 to get 5.

$$\sqrt{\frac{25\left(\frac{5}{20}+\frac{1}{10}\right)\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

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

$$\sqrt{\frac{25\left(\frac{1}{4}+\frac{1}{10}\right)\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Least common multiple of 4 and 10 is 20. Convert \frac{1}{4} and \frac{1}{10} to fractions with denominator 20.

$$\sqrt{\frac{25\left(\frac{5}{20}+\frac{2}{20}\right)\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Since \frac{5}{20} and \frac{2}{20} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{25\times \frac{5+2}{20}\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Add 5 and 2 to get 7.

$$\sqrt{\frac{25\times \frac{7}{20}\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Express 25\times \frac{7}{20} as a single fraction.

$$\sqrt{\frac{\frac{25\times 7}{20}\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Multiply 25 and 7 to get 175.

$$\sqrt{\frac{\frac{175}{20}\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Reduce the fraction \frac{175}{20} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{\frac{35}{4}\times \frac{4}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Multiply \frac{35}{4} times \frac{4}{5} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{35\times 4}{4\times 5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Cancel out 4 in both numerator and denominator.

$$\sqrt{\frac{\frac{35}{5}\times \frac{1}{14}}{\frac{18}{32}}}$$

Divide 35 by 5 to get 7.

$$\sqrt{\frac{7\times \frac{1}{14}}{\frac{18}{32}}}$$

Multiply 7 and \frac{1}{14} to get \frac{7}{14}.

$$\sqrt{\frac{\frac{7}{14}}{\frac{18}{32}}}$$

Reduce the fraction \frac{7}{14} to lowest terms by extracting and canceling out 7.

$$\sqrt{\frac{\frac{1}{2}}{\frac{18}{32}}}$$

Reduce the fraction \frac{18}{32} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{\frac{1}{2}}{\frac{9}{16}}}$$

Divide \frac{1}{2} by \frac{9}{16} by multiplying \frac{1}{2} by the reciprocal of \frac{9}{16}.

$$\sqrt{\frac{1}{2}\times \frac{16}{9}}$$

Multiply \frac{1}{2} times \frac{16}{9} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{1\times 16}{2\times 9}}$$

Do the multiplications in the fraction \frac{1\times 16}{2\times 9}.

$$\sqrt{\frac{16}{18}}$$

Reduce the fraction \frac{16}{18} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{8}{9}}$$

Rewrite the square root of the division \sqrt{\frac{8}{9}} as the division of square roots \frac{\sqrt{8}}{\sqrt{9}}.

$$\frac{\sqrt{8}}{\sqrt{9}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{2\sqrt{2}}{\sqrt{9}}$$

Calculate the square root of 9 and get 3.

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