Solve sqrt{{1/4+13/34:[3/4:(4/6+frac{5}{4}*frac{3}{5})+1]+frac{5}{2}}*frac{2^7}{16}}

$$\sqrt { \{ \frac { 1 } { 4 } + \frac { 13 } { 34 } : [ \frac { 3 } { 4 } : ( \frac { 4 } { 6 } + \frac { 5 } { 4 } \cdot \frac { 3 } { 5 } ) + 1 ] + \frac { 5 } { 2 } \} \cdot \frac { 2 ^ { 7 } } { 16 } }$$

Evaluate

$$2\sqrt{6}\approx 4.898979486$$

Solution Steps

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

$$\sqrt{\left(\frac{1}{4}+\frac{\frac{13}{34}}{\frac{\frac{3}{4}}{\frac{2}{3}+\frac{5}{4}\times \frac{3}{5}}+1}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Multiply \frac{5}{4} times \frac{3}{5} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\left(\frac{1}{4}+\frac{\frac{13}{34}}{\frac{\frac{3}{4}}{\frac{2}{3}+\frac{5\times 3}{4\times 5}}+1}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Cancel out 5 in both numerator and denominator.

$$\sqrt{\left(\frac{1}{4}+\frac{\frac{13}{34}}{\frac{\frac{3}{4}}{\frac{2}{3}+\frac{3}{4}}+1}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Least common multiple of 3 and 4 is 12. Convert \frac{2}{3} and \frac{3}{4} to fractions with denominator 12.

$$\sqrt{\left(\frac{1}{4}+\frac{\frac{13}{34}}{\frac{\frac{3}{4}}{\frac{8}{12}+\frac{9}{12}}+1}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Since \frac{8}{12} and \frac{9}{12} have the same denominator, add them by adding their numerators.

$$\sqrt{\left(\frac{1}{4}+\frac{\frac{13}{34}}{\frac{\frac{3}{4}}{\frac{8+9}{12}}+1}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Add 8 and 9 to get 17.

$$\sqrt{\left(\frac{1}{4}+\frac{\frac{13}{34}}{\frac{\frac{3}{4}}{\frac{17}{12}}+1}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Divide \frac{3}{4} by \frac{17}{12} by multiplying \frac{3}{4} by the reciprocal of \frac{17}{12}.

$$\sqrt{\left(\frac{1}{4}+\frac{\frac{13}{34}}{\frac{3}{4}\times \frac{12}{17}+1}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Multiply \frac{3}{4} times \frac{12}{17} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\left(\frac{1}{4}+\frac{\frac{13}{34}}{\frac{3\times 12}{4\times 17}+1}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Do the multiplications in the fraction \frac{3\times 12}{4\times 17}.

$$\sqrt{\left(\frac{1}{4}+\frac{\frac{13}{34}}{\frac{36}{68}+1}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Reduce the fraction \frac{36}{68} to lowest terms by extracting and canceling out 4.

$$\sqrt{\left(\frac{1}{4}+\frac{\frac{13}{34}}{\frac{9}{17}+1}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Convert 1 to fraction \frac{17}{17}.

$$\sqrt{\left(\frac{1}{4}+\frac{\frac{13}{34}}{\frac{9}{17}+\frac{17}{17}}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Since \frac{9}{17} and \frac{17}{17} have the same denominator, add them by adding their numerators.

$$\sqrt{\left(\frac{1}{4}+\frac{\frac{13}{34}}{\frac{9+17}{17}}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Add 9 and 17 to get 26.

$$\sqrt{\left(\frac{1}{4}+\frac{\frac{13}{34}}{\frac{26}{17}}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Divide \frac{13}{34} by \frac{26}{17} by multiplying \frac{13}{34} by the reciprocal of \frac{26}{17}.

$$\sqrt{\left(\frac{1}{4}+\frac{13}{34}\times \frac{17}{26}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Multiply \frac{13}{34} times \frac{17}{26} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\left(\frac{1}{4}+\frac{13\times 17}{34\times 26}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Do the multiplications in the fraction \frac{13\times 17}{34\times 26}.

$$\sqrt{\left(\frac{1}{4}+\frac{221}{884}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Reduce the fraction \frac{221}{884} to lowest terms by extracting and canceling out 221.

$$\sqrt{\left(\frac{1}{4}+\frac{1}{4}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

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

$$\sqrt{\left(\frac{1+1}{4}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Add 1 and 1 to get 2.

$$\sqrt{\left(\frac{2}{4}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\sqrt{\left(\frac{1}{2}+\frac{5}{2}\right)\times \frac{2^{7}}{16}}$$

Since \frac{1}{2} and \frac{5}{2} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1+5}{2}\times \frac{2^{7}}{16}}$$

Add 1 and 5 to get 6.

$$\sqrt{\frac{6}{2}\times \frac{2^{7}}{16}}$$

Divide 6 by 2 to get 3.

$$\sqrt{3\times \frac{2^{7}}{16}}$$

Calculate 2 to the power of 7 and get 128.

$$\sqrt{3\times \frac{128}{16}}$$

Divide 128 by 16 to get 8.

$$\sqrt{3\times 8}$$

Multiply 3 and 8 to get 24.

$$\sqrt{24}$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$2\sqrt{6}$$