Solve sqrt{(2+1/2*11/10)*((2/3)^2:(5+2/3)):(frac{7}{45}:7+frac{1}{3})*2^4}

$$\sqrt { ( 2 + \frac { 1 } { 2 } \cdot \frac { 11 } { 10 } ) \cdot ( ( \frac { 2 } { 3 } ) ^ { 2 } : ( 5 + \frac { 2 } { 3 } ) ) : ( \frac { 7 } { 45 } : 7 + \frac { 1 } { 3 } ) \cdot 2 ^ { 4 } }$$

Evaluate

$$3$$

Solution Steps

Multiply \frac{1}{2} times \frac{11}{10} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\left(2+\frac{1\times 11}{2\times 10}\right)\times \frac{\left(\frac{2}{3}\right)^{2}}{5+\frac{2}{3}}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Do the multiplications in the fraction \frac{1\times 11}{2\times 10}.

$$\sqrt{\frac{\left(2+\frac{11}{20}\right)\times \frac{\left(\frac{2}{3}\right)^{2}}{5+\frac{2}{3}}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Convert 2 to fraction \frac{40}{20}.

$$\sqrt{\frac{\left(\frac{40}{20}+\frac{11}{20}\right)\times \frac{\left(\frac{2}{3}\right)^{2}}{5+\frac{2}{3}}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Since \frac{40}{20} and \frac{11}{20} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{40+11}{20}\times \frac{\left(\frac{2}{3}\right)^{2}}{5+\frac{2}{3}}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Add 40 and 11 to get 51.

$$\sqrt{\frac{\frac{51}{20}\times \frac{\left(\frac{2}{3}\right)^{2}}{5+\frac{2}{3}}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Calculate \frac{2}{3} to the power of 2 and get \frac{4}{9}.

$$\sqrt{\frac{\frac{51}{20}\times \frac{\frac{4}{9}}{5+\frac{2}{3}}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Convert 5 to fraction \frac{15}{3}.

$$\sqrt{\frac{\frac{51}{20}\times \frac{\frac{4}{9}}{\frac{15}{3}+\frac{2}{3}}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Since \frac{15}{3} and \frac{2}{3} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{51}{20}\times \frac{\frac{4}{9}}{\frac{15+2}{3}}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Add 15 and 2 to get 17.

$$\sqrt{\frac{\frac{51}{20}\times \frac{\frac{4}{9}}{\frac{17}{3}}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Divide \frac{4}{9} by \frac{17}{3} by multiplying \frac{4}{9} by the reciprocal of \frac{17}{3}.

$$\sqrt{\frac{\frac{51}{20}\times \frac{4}{9}\times \frac{3}{17}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Multiply \frac{4}{9} times \frac{3}{17} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{51}{20}\times \frac{4\times 3}{9\times 17}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Do the multiplications in the fraction \frac{4\times 3}{9\times 17}.

$$\sqrt{\frac{\frac{51}{20}\times \frac{12}{153}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Reduce the fraction \frac{12}{153} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{\frac{51}{20}\times \frac{4}{51}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Multiply \frac{51}{20} times \frac{4}{51} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{51\times 4}{20\times 51}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Cancel out 51 in both numerator and denominator.

$$\sqrt{\frac{\frac{4}{20}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Reduce the fraction \frac{4}{20} to lowest terms by extracting and canceling out 4.

$$\sqrt{\frac{\frac{1}{5}}{\frac{\frac{7}{45}}{7}+\frac{1}{3}}\times 2^{4}}$$

Express \frac{\frac{7}{45}}{7} as a single fraction.

$$\sqrt{\frac{\frac{1}{5}}{\frac{7}{45\times 7}+\frac{1}{3}}\times 2^{4}}$$

Cancel out 7 in both numerator and denominator.

$$\sqrt{\frac{\frac{1}{5}}{\frac{1}{45}+\frac{1}{3}}\times 2^{4}}$$

Least common multiple of 45 and 3 is 45. Convert \frac{1}{45} and \frac{1}{3} to fractions with denominator 45.

$$\sqrt{\frac{\frac{1}{5}}{\frac{1}{45}+\frac{15}{45}}\times 2^{4}}$$

Since \frac{1}{45} and \frac{15}{45} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{1}{5}}{\frac{1+15}{45}}\times 2^{4}}$$

Add 1 and 15 to get 16.

$$\sqrt{\frac{\frac{1}{5}}{\frac{16}{45}}\times 2^{4}}$$

Divide \frac{1}{5} by \frac{16}{45} by multiplying \frac{1}{5} by the reciprocal of \frac{16}{45}.

$$\sqrt{\frac{1}{5}\times \frac{45}{16}\times 2^{4}}$$

Multiply \frac{1}{5} times \frac{45}{16} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{1\times 45}{5\times 16}\times 2^{4}}$$

Do the multiplications in the fraction \frac{1\times 45}{5\times 16}.

$$\sqrt{\frac{45}{80}\times 2^{4}}$$

Reduce the fraction \frac{45}{80} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{9}{16}\times 2^{4}}$$

Calculate 2 to the power of 4 and get 16.

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

Cancel out 16 and 16.

$$\sqrt{9}$$

Calculate the square root of 9 and get 3.

$$3$$

Factor

$$3$$