Solve sqrt{(1+2/3)*(1-2/5)+(1-3/7)*(3/4+frac{1}{2})}=

$$\sqrt { ( 1 + \frac { 2 } { 3 } ) * ( 1 - \frac { 2 } { 5 } ) + ( 1 - \frac { 3 } { 7 } ) * ( \frac { 3 } { 4 } + \frac { 1 } { 2 } ) } =$$

Evaluate

$$\frac{2\sqrt{21}}{7}\approx 1.309307341$$

Solution Steps

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

$$\sqrt{\left(\frac{3}{3}+\frac{2}{3}\right)\left(1-\frac{2}{5}\right)+\left(1-\frac{3}{7}\right)\left(\frac{3}{4}+\frac{1}{2}\right)}$$

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

$$\sqrt{\frac{3+2}{3}\left(1-\frac{2}{5}\right)+\left(1-\frac{3}{7}\right)\left(\frac{3}{4}+\frac{1}{2}\right)}$$

Add 3 and 2 to get 5.

$$\sqrt{\frac{5}{3}\left(1-\frac{2}{5}\right)+\left(1-\frac{3}{7}\right)\left(\frac{3}{4}+\frac{1}{2}\right)}$$

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

$$\sqrt{\frac{5}{3}\left(\frac{5}{5}-\frac{2}{5}\right)+\left(1-\frac{3}{7}\right)\left(\frac{3}{4}+\frac{1}{2}\right)}$$

Since \frac{5}{5} and \frac{2}{5} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{5}{3}\times \frac{5-2}{5}+\left(1-\frac{3}{7}\right)\left(\frac{3}{4}+\frac{1}{2}\right)}$$

Subtract 2 from 5 to get 3.

$$\sqrt{\frac{5}{3}\times \frac{3}{5}+\left(1-\frac{3}{7}\right)\left(\frac{3}{4}+\frac{1}{2}\right)}$$

Cancel out \frac{5}{3} and its reciprocal \frac{3}{5}.

$$\sqrt{1+\left(1-\frac{3}{7}\right)\left(\frac{3}{4}+\frac{1}{2}\right)}$$

Convert 1 to fraction \frac{7}{7}.

$$\sqrt{1+\left(\frac{7}{7}-\frac{3}{7}\right)\left(\frac{3}{4}+\frac{1}{2}\right)}$$

Since \frac{7}{7} and \frac{3}{7} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{1+\frac{7-3}{7}\left(\frac{3}{4}+\frac{1}{2}\right)}$$

Subtract 3 from 7 to get 4.

$$\sqrt{1+\frac{4}{7}\left(\frac{3}{4}+\frac{1}{2}\right)}$$

Least common multiple of 4 and 2 is 4. Convert \frac{3}{4} and \frac{1}{2} to fractions with denominator 4.

$$\sqrt{1+\frac{4}{7}\left(\frac{3}{4}+\frac{2}{4}\right)}$$

Since \frac{3}{4} and \frac{2}{4} have the same denominator, add them by adding their numerators.

$$\sqrt{1+\frac{4}{7}\times \frac{3+2}{4}}$$

Add 3 and 2 to get 5.

$$\sqrt{1+\frac{4}{7}\times \frac{5}{4}}$$

Multiply \frac{4}{7} times \frac{5}{4} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{1+\frac{4\times 5}{7\times 4}}$$

Cancel out 4 in both numerator and denominator.

$$\sqrt{1+\frac{5}{7}}$$

Convert 1 to fraction \frac{7}{7}.

$$\sqrt{\frac{7}{7}+\frac{5}{7}}$$

Since \frac{7}{7} and \frac{5}{7} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{7+5}{7}}$$

Add 7 and 5 to get 12.

$$\sqrt{\frac{12}{7}}$$

Rewrite the square root of the division \sqrt{\frac{12}{7}} as the division of square roots \frac{\sqrt{12}}{\sqrt{7}}.

$$\frac{\sqrt{12}}{\sqrt{7}}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

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

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

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

The square of \sqrt{7} is 7.

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

To multiply \sqrt{3} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{2\sqrt{21}}{7}$$