Solve 22sqrt{[(1/2)^2+(1-1/3)-1/2]*(1-1/5)*(2+frac{2}{5}):20}

$$22 \sqrt { [ ( \frac { 1 } { 2 } ) ^ { 2 } + ( 1 - \frac { 1 } { 3 } ) - \frac { 1 } { 2 } ] \times ( 1 - \frac { 1 } { 5 } ) \times ( 2 + \frac { 2 } { 5 } ) : 20 }$$

Evaluate

$$\frac{22}{5}=4.4$$

Solution Steps

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$22\sqrt{\frac{\left(\frac{1}{4}+1-\frac{1}{3}-\frac{1}{2}\right)\left(1-\frac{1}{5}\right)\left(2+\frac{2}{5}\right)}{20}}$$

Convert 1 to fraction \frac{4}{4}.

$$22\sqrt{\frac{\left(\frac{1}{4}+\frac{4}{4}-\frac{1}{3}-\frac{1}{2}\right)\left(1-\frac{1}{5}\right)\left(2+\frac{2}{5}\right)}{20}}$$

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

$$22\sqrt{\frac{\left(\frac{1+4}{4}-\frac{1}{3}-\frac{1}{2}\right)\left(1-\frac{1}{5}\right)\left(2+\frac{2}{5}\right)}{20}}$$

Add 1 and 4 to get 5.

$$22\sqrt{\frac{\left(\frac{5}{4}-\frac{1}{3}-\frac{1}{2}\right)\left(1-\frac{1}{5}\right)\left(2+\frac{2}{5}\right)}{20}}$$

Least common multiple of 4 and 3 is 12. Convert \frac{5}{4} and \frac{1}{3} to fractions with denominator 12.

$$22\sqrt{\frac{\left(\frac{15}{12}-\frac{4}{12}-\frac{1}{2}\right)\left(1-\frac{1}{5}\right)\left(2+\frac{2}{5}\right)}{20}}$$

Since \frac{15}{12} and \frac{4}{12} have the same denominator, subtract them by subtracting their numerators.

$$22\sqrt{\frac{\left(\frac{15-4}{12}-\frac{1}{2}\right)\left(1-\frac{1}{5}\right)\left(2+\frac{2}{5}\right)}{20}}$$

Subtract 4 from 15 to get 11.

$$22\sqrt{\frac{\left(\frac{11}{12}-\frac{1}{2}\right)\left(1-\frac{1}{5}\right)\left(2+\frac{2}{5}\right)}{20}}$$

Least common multiple of 12 and 2 is 12. Convert \frac{11}{12} and \frac{1}{2} to fractions with denominator 12.

$$22\sqrt{\frac{\left(\frac{11}{12}-\frac{6}{12}\right)\left(1-\frac{1}{5}\right)\left(2+\frac{2}{5}\right)}{20}}$$

Since \frac{11}{12} and \frac{6}{12} have the same denominator, subtract them by subtracting their numerators.

$$22\sqrt{\frac{\frac{11-6}{12}\left(1-\frac{1}{5}\right)\left(2+\frac{2}{5}\right)}{20}}$$

Subtract 6 from 11 to get 5.

$$22\sqrt{\frac{\frac{5}{12}\left(1-\frac{1}{5}\right)\left(2+\frac{2}{5}\right)}{20}}$$

Convert 1 to fraction \frac{5}{5}.

$$22\sqrt{\frac{\frac{5}{12}\left(\frac{5}{5}-\frac{1}{5}\right)\left(2+\frac{2}{5}\right)}{20}}$$

Since \frac{5}{5} and \frac{1}{5} have the same denominator, subtract them by subtracting their numerators.

$$22\sqrt{\frac{\frac{5}{12}\times \frac{5-1}{5}\left(2+\frac{2}{5}\right)}{20}}$$

Subtract 1 from 5 to get 4.

$$22\sqrt{\frac{\frac{5}{12}\times \frac{4}{5}\left(2+\frac{2}{5}\right)}{20}}$$

Multiply \frac{5}{12} times \frac{4}{5} by multiplying numerator times numerator and denominator times denominator.

$$22\sqrt{\frac{\frac{5\times 4}{12\times 5}\left(2+\frac{2}{5}\right)}{20}}$$

Cancel out 5 in both numerator and denominator.

$$22\sqrt{\frac{\frac{4}{12}\left(2+\frac{2}{5}\right)}{20}}$$

Reduce the fraction \frac{4}{12} to lowest terms by extracting and canceling out 4.

$$22\sqrt{\frac{\frac{1}{3}\left(2+\frac{2}{5}\right)}{20}}$$

Convert 2 to fraction \frac{10}{5}.

$$22\sqrt{\frac{\frac{1}{3}\left(\frac{10}{5}+\frac{2}{5}\right)}{20}}$$

Since \frac{10}{5} and \frac{2}{5} have the same denominator, add them by adding their numerators.

$$22\sqrt{\frac{\frac{1}{3}\times \frac{10+2}{5}}{20}}$$

Add 10 and 2 to get 12.

$$22\sqrt{\frac{\frac{1}{3}\times \frac{12}{5}}{20}}$$

Multiply \frac{1}{3} times \frac{12}{5} by multiplying numerator times numerator and denominator times denominator.

$$22\sqrt{\frac{\frac{1\times 12}{3\times 5}}{20}}$$

Do the multiplications in the fraction \frac{1\times 12}{3\times 5}.

$$22\sqrt{\frac{\frac{12}{15}}{20}}$$

Reduce the fraction \frac{12}{15} to lowest terms by extracting and canceling out 3.

$$22\sqrt{\frac{\frac{4}{5}}{20}}$$

Express \frac{\frac{4}{5}}{20} as a single fraction.

$$22\sqrt{\frac{4}{5\times 20}}$$

Multiply 5 and 20 to get 100.

$$22\sqrt{\frac{4}{100}}$$

Reduce the fraction \frac{4}{100} to lowest terms by extracting and canceling out 4.

$$22\sqrt{\frac{1}{25}}$$

Rewrite the square root of the division \frac{1}{25} as the division of square roots \frac{\sqrt{1}}{\sqrt{25}}. Take the square root of both numerator and denominator.

$$22\times \frac{1}{5}$$

Multiply 22 and \frac{1}{5} to get \frac{22}{5}.

$$\frac{22}{5}$$

Factor

$$\frac{2 \cdot 11}{5} = 4\frac{2}{5} = 4.4$$