Solve sqrt{(7/9+2/3-5/27)-{5/9-[frac{2}{3}-(frac{4}{27}+frac{4}{9})]}}

$$\sqrt { ( \frac { 7 } { 9 } + \frac { 2 } { 3 } - \frac { 5 } { 27 } ) - \{ \frac { 5 } { 9 } - [ \frac { 2 } { 3 } - ( \frac { 4 } { 27 } + \frac { 4 } { 9 } ) ] \} }$$

Evaluate

$$\frac{\sqrt{7}}{3}\approx 0.881917104$$

Solution Steps

Least common multiple of 9 and 3 is 9. Convert \frac{7}{9} and \frac{2}{3} to fractions with denominator 9.

$$\sqrt{\frac{7}{9}+\frac{6}{9}-\frac{5}{27}-\left(\frac{5}{9}-\left(\frac{2}{3}-\left(\frac{4}{27}+\frac{4}{9}\right)\right)\right)}$$

Since \frac{7}{9} and \frac{6}{9} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{7+6}{9}-\frac{5}{27}-\left(\frac{5}{9}-\left(\frac{2}{3}-\left(\frac{4}{27}+\frac{4}{9}\right)\right)\right)}$$

Add 7 and 6 to get 13.

$$\sqrt{\frac{13}{9}-\frac{5}{27}-\left(\frac{5}{9}-\left(\frac{2}{3}-\left(\frac{4}{27}+\frac{4}{9}\right)\right)\right)}$$

Least common multiple of 9 and 27 is 27. Convert \frac{13}{9} and \frac{5}{27} to fractions with denominator 27.

$$\sqrt{\frac{39}{27}-\frac{5}{27}-\left(\frac{5}{9}-\left(\frac{2}{3}-\left(\frac{4}{27}+\frac{4}{9}\right)\right)\right)}$$

Since \frac{39}{27} and \frac{5}{27} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{39-5}{27}-\left(\frac{5}{9}-\left(\frac{2}{3}-\left(\frac{4}{27}+\frac{4}{9}\right)\right)\right)}$$

Subtract 5 from 39 to get 34.

$$\sqrt{\frac{34}{27}-\left(\frac{5}{9}-\left(\frac{2}{3}-\left(\frac{4}{27}+\frac{4}{9}\right)\right)\right)}$$

Least common multiple of 27 and 9 is 27. Convert \frac{4}{27} and \frac{4}{9} to fractions with denominator 27.

$$\sqrt{\frac{34}{27}-\left(\frac{5}{9}-\left(\frac{2}{3}-\left(\frac{4}{27}+\frac{12}{27}\right)\right)\right)}$$

Since \frac{4}{27} and \frac{12}{27} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{34}{27}-\left(\frac{5}{9}-\left(\frac{2}{3}-\frac{4+12}{27}\right)\right)}$$

Add 4 and 12 to get 16.

$$\sqrt{\frac{34}{27}-\left(\frac{5}{9}-\left(\frac{2}{3}-\frac{16}{27}\right)\right)}$$

Least common multiple of 3 and 27 is 27. Convert \frac{2}{3} and \frac{16}{27} to fractions with denominator 27.

$$\sqrt{\frac{34}{27}-\left(\frac{5}{9}-\left(\frac{18}{27}-\frac{16}{27}\right)\right)}$$

Since \frac{18}{27} and \frac{16}{27} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{34}{27}-\left(\frac{5}{9}-\frac{18-16}{27}\right)}$$

Subtract 16 from 18 to get 2.

$$\sqrt{\frac{34}{27}-\left(\frac{5}{9}-\frac{2}{27}\right)}$$

Least common multiple of 9 and 27 is 27. Convert \frac{5}{9} and \frac{2}{27} to fractions with denominator 27.

$$\sqrt{\frac{34}{27}-\left(\frac{15}{27}-\frac{2}{27}\right)}$$

Since \frac{15}{27} and \frac{2}{27} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{34}{27}-\frac{15-2}{27}}$$

Subtract 2 from 15 to get 13.

$$\sqrt{\frac{34}{27}-\frac{13}{27}}$$

Since \frac{34}{27} and \frac{13}{27} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{34-13}{27}}$$

Subtract 13 from 34 to get 21.

$$\sqrt{\frac{21}{27}}$$

Reduce the fraction \frac{21}{27} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{7}{9}}$$

Rewrite the square root of the division \sqrt{\frac{7}{9}} as the division of square roots \frac{\sqrt{7}}{\sqrt{9}}.

$$\frac{\sqrt{7}}{\sqrt{9}}$$

Calculate the square root of 9 and get 3.

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