Solve sqrt{2/1-{[(12/11+4/3)-(3/5+frac{1}{3})]*frac{1}{41}*frac{8}{5}}}

$$\sqrt { \frac { 2 } { 1 } - \{ [ ( \frac { 12 } { 11 } + \frac { 4 } { 3 } ) - ( \frac { 3 } { 5 } + \frac { 1 } { 3 } ) ] \cdot \frac { 1 } { 41 } \cdot \frac { 8 } { 5 } \} }$$

Evaluate

$$\frac{\sqrt{5874}}{55}\approx 1.393491364$$

Solution Steps

Anything divided by one gives itself.

$$\sqrt{2-\left(\frac{12}{11}+\frac{4}{3}-\left(\frac{3}{5}+\frac{1}{3}\right)\right)\times \frac{1}{41}\times \frac{8}{5}}$$

Least common multiple of 11 and 3 is 33. Convert \frac{12}{11} and \frac{4}{3} to fractions with denominator 33.

$$\sqrt{2-\left(\frac{36}{33}+\frac{44}{33}-\left(\frac{3}{5}+\frac{1}{3}\right)\right)\times \frac{1}{41}\times \frac{8}{5}}$$

Since \frac{36}{33} and \frac{44}{33} have the same denominator, add them by adding their numerators.

$$\sqrt{2-\left(\frac{36+44}{33}-\left(\frac{3}{5}+\frac{1}{3}\right)\right)\times \frac{1}{41}\times \frac{8}{5}}$$

Add 36 and 44 to get 80.

$$\sqrt{2-\left(\frac{80}{33}-\left(\frac{3}{5}+\frac{1}{3}\right)\right)\times \frac{1}{41}\times \frac{8}{5}}$$

Least common multiple of 5 and 3 is 15. Convert \frac{3}{5} and \frac{1}{3} to fractions with denominator 15.

$$\sqrt{2-\left(\frac{80}{33}-\left(\frac{9}{15}+\frac{5}{15}\right)\right)\times \frac{1}{41}\times \frac{8}{5}}$$

Since \frac{9}{15} and \frac{5}{15} have the same denominator, add them by adding their numerators.

$$\sqrt{2-\left(\frac{80}{33}-\frac{9+5}{15}\right)\times \frac{1}{41}\times \frac{8}{5}}$$

Add 9 and 5 to get 14.

$$\sqrt{2-\left(\frac{80}{33}-\frac{14}{15}\right)\times \frac{1}{41}\times \frac{8}{5}}$$

Least common multiple of 33 and 15 is 165. Convert \frac{80}{33} and \frac{14}{15} to fractions with denominator 165.

$$\sqrt{2-\left(\frac{400}{165}-\frac{154}{165}\right)\times \frac{1}{41}\times \frac{8}{5}}$$

Since \frac{400}{165} and \frac{154}{165} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{2-\frac{400-154}{165}\times \frac{1}{41}\times \frac{8}{5}}$$

Subtract 154 from 400 to get 246.

$$\sqrt{2-\frac{246}{165}\times \frac{1}{41}\times \frac{8}{5}}$$

Reduce the fraction \frac{246}{165} to lowest terms by extracting and canceling out 3.

$$\sqrt{2-\frac{82}{55}\times \frac{1}{41}\times \frac{8}{5}}$$

Multiply \frac{82}{55} times \frac{1}{41} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{2-\frac{82\times 1}{55\times 41}\times \frac{8}{5}}$$

Do the multiplications in the fraction \frac{82\times 1}{55\times 41}.

$$\sqrt{2-\frac{82}{2255}\times \frac{8}{5}}$$

Reduce the fraction \frac{82}{2255} to lowest terms by extracting and canceling out 41.

$$\sqrt{2-\frac{2}{55}\times \frac{8}{5}}$$

Multiply \frac{2}{55} times \frac{8}{5} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{2-\frac{2\times 8}{55\times 5}}$$

Do the multiplications in the fraction \frac{2\times 8}{55\times 5}.

$$\sqrt{2-\frac{16}{275}}$$

Convert 2 to fraction \frac{550}{275}.

$$\sqrt{\frac{550}{275}-\frac{16}{275}}$$

Since \frac{550}{275} and \frac{16}{275} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{550-16}{275}}$$

Subtract 16 from 550 to get 534.

$$\sqrt{\frac{534}{275}}$$

Rewrite the square root of the division \sqrt{\frac{534}{275}} as the division of square roots \frac{\sqrt{534}}{\sqrt{275}}.

$$\frac{\sqrt{534}}{\sqrt{275}}$$

Factor 275=5^{2}\times 11. Rewrite the square root of the product \sqrt{5^{2}\times 11} as the product of square roots \sqrt{5^{2}}\sqrt{11}. Take the square root of 5^{2}.

$$\frac{\sqrt{534}}{5\sqrt{11}}$$

Rationalize the denominator of \frac{\sqrt{534}}{5\sqrt{11}} by multiplying numerator and denominator by \sqrt{11}.

$$\frac{\sqrt{534}\sqrt{11}}{5\left(\sqrt{11}\right)^{2}}$$

The square of \sqrt{11} is 11.

$$\frac{\sqrt{534}\sqrt{11}}{5\times 11}$$

To multiply \sqrt{534} and \sqrt{11}, multiply the numbers under the square root.

$$\frac{\sqrt{5874}}{5\times 11}$$

Multiply 5 and 11 to get 55.

$$\frac{\sqrt{5874}}{55}$$