Solve 1/3+sqrt{7/9*sqrt{sqrt{1/2*1/4*frac{32}{49}}+frac{2}{49}}}

$$\frac { 1 } { 3 } + \sqrt { \frac { 7 } { 9 } \times \sqrt { \sqrt { \frac { 1 } { 2 } \times \frac { 1 } { 4 } \times \frac { 32 } { 49 } } + \frac { 2 } { 49 } } }$$

Evaluate

$$1$$

Solution Steps

Multiply \frac{1}{2} times \frac{1}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{3}+\sqrt{\frac{7}{9}\sqrt{\sqrt{\frac{1\times 1}{2\times 4}\times \frac{32}{49}}+\frac{2}{49}}}$$

Do the multiplications in the fraction \frac{1\times 1}{2\times 4}.

$$\frac{1}{3}+\sqrt{\frac{7}{9}\sqrt{\sqrt{\frac{1}{8}\times \frac{32}{49}}+\frac{2}{49}}}$$

Multiply \frac{1}{8} times \frac{32}{49} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{3}+\sqrt{\frac{7}{9}\sqrt{\sqrt{\frac{1\times 32}{8\times 49}}+\frac{2}{49}}}$$

Do the multiplications in the fraction \frac{1\times 32}{8\times 49}.

$$\frac{1}{3}+\sqrt{\frac{7}{9}\sqrt{\sqrt{\frac{32}{392}}+\frac{2}{49}}}$$

Reduce the fraction \frac{32}{392} to lowest terms by extracting and canceling out 8.

$$\frac{1}{3}+\sqrt{\frac{7}{9}\sqrt{\sqrt{\frac{4}{49}}+\frac{2}{49}}}$$

Rewrite the square root of the division \frac{4}{49} as the division of square roots \frac{\sqrt{4}}{\sqrt{49}}. Take the square root of both numerator and denominator.

$$\frac{1}{3}+\sqrt{\frac{7}{9}\sqrt{\frac{2}{7}+\frac{2}{49}}}$$

Least common multiple of 7 and 49 is 49. Convert \frac{2}{7} and \frac{2}{49} to fractions with denominator 49.

$$\frac{1}{3}+\sqrt{\frac{7}{9}\sqrt{\frac{14}{49}+\frac{2}{49}}}$$

Since \frac{14}{49} and \frac{2}{49} have the same denominator, add them by adding their numerators.

$$\frac{1}{3}+\sqrt{\frac{7}{9}\sqrt{\frac{14+2}{49}}}$$

Add 14 and 2 to get 16.

$$\frac{1}{3}+\sqrt{\frac{7}{9}\sqrt{\frac{16}{49}}}$$

Rewrite the square root of the division \frac{16}{49} as the division of square roots \frac{\sqrt{16}}{\sqrt{49}}. Take the square root of both numerator and denominator.

$$\frac{1}{3}+\sqrt{\frac{7}{9}\times \frac{4}{7}}$$

Multiply \frac{7}{9} times \frac{4}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{3}+\sqrt{\frac{7\times 4}{9\times 7}}$$

Cancel out 7 in both numerator and denominator.

$$\frac{1}{3}+\sqrt{\frac{4}{9}}$$

Rewrite the square root of the division \frac{4}{9} as the division of square roots \frac{\sqrt{4}}{\sqrt{9}}. Take the square root of both numerator and denominator.

$$\frac{1}{3}+\frac{2}{3}$$

Since \frac{1}{3} and \frac{2}{3} have the same denominator, add them by adding their numerators.

$$\frac{1+2}{3}$$

Add 1 and 2 to get 3.

$$\frac{3}{3}$$

Divide 3 by 3 to get 1.

$$1$$

Factor

$$1$$