Solve sqrt{[(26/3+1/4-7)*(1-11/15)+8/4]:(frac{1}{4}+1+frac{1}{2}*frac{7}{5})}

$$\sqrt { [ ( \frac { 26 } { 3 } + \frac { 1 } { 4 } - 7 ) \times ( 1 - \frac { 11 } { 15 } ) + \frac { 8 } { 4 } ] : ( \frac { 1 } { 4 } + 1 + \frac { 1 } { 2 } \cdot \frac { 7 } { 5 } ) }$$

Evaluate

$$\frac{2\sqrt{4407}}{117}\approx 1.134790416$$

Solution Steps

Least common multiple of 3 and 4 is 12. Convert \frac{26}{3} and \frac{1}{4} to fractions with denominator 12.

$$\sqrt{\frac{\left(\frac{104}{12}+\frac{3}{12}-7\right)\left(1-\frac{11}{15}\right)+\frac{8}{4}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Since \frac{104}{12} and \frac{3}{12} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\left(\frac{104+3}{12}-7\right)\left(1-\frac{11}{15}\right)+\frac{8}{4}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Add 104 and 3 to get 107.

$$\sqrt{\frac{\left(\frac{107}{12}-7\right)\left(1-\frac{11}{15}\right)+\frac{8}{4}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Convert 7 to fraction \frac{84}{12}.

$$\sqrt{\frac{\left(\frac{107}{12}-\frac{84}{12}\right)\left(1-\frac{11}{15}\right)+\frac{8}{4}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Since \frac{107}{12} and \frac{84}{12} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{107-84}{12}\left(1-\frac{11}{15}\right)+\frac{8}{4}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Subtract 84 from 107 to get 23.

$$\sqrt{\frac{\frac{23}{12}\left(1-\frac{11}{15}\right)+\frac{8}{4}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Convert 1 to fraction \frac{15}{15}.

$$\sqrt{\frac{\frac{23}{12}\left(\frac{15}{15}-\frac{11}{15}\right)+\frac{8}{4}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Since \frac{15}{15} and \frac{11}{15} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{23}{12}\times \frac{15-11}{15}+\frac{8}{4}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Subtract 11 from 15 to get 4.

$$\sqrt{\frac{\frac{23}{12}\times \frac{4}{15}+\frac{8}{4}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Multiply \frac{23}{12} times \frac{4}{15} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{23\times 4}{12\times 15}+\frac{8}{4}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Do the multiplications in the fraction \frac{23\times 4}{12\times 15}.

$$\sqrt{\frac{\frac{92}{180}+\frac{8}{4}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Reduce the fraction \frac{92}{180} to lowest terms by extracting and canceling out 4.

$$\sqrt{\frac{\frac{23}{45}+\frac{8}{4}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Divide 8 by 4 to get 2.

$$\sqrt{\frac{\frac{23}{45}+2}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Convert 2 to fraction \frac{90}{45}.

$$\sqrt{\frac{\frac{23}{45}+\frac{90}{45}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Since \frac{23}{45} and \frac{90}{45} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{23+90}{45}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Add 23 and 90 to get 113.

$$\sqrt{\frac{\frac{113}{45}}{\frac{1}{4}+1+\frac{1}{2}\times \frac{7}{5}}}$$

Convert 1 to fraction \frac{4}{4}.

$$\sqrt{\frac{\frac{113}{45}}{\frac{1}{4}+\frac{4}{4}+\frac{1}{2}\times \frac{7}{5}}}$$

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

$$\sqrt{\frac{\frac{113}{45}}{\frac{1+4}{4}+\frac{1}{2}\times \frac{7}{5}}}$$

Add 1 and 4 to get 5.

$$\sqrt{\frac{\frac{113}{45}}{\frac{5}{4}+\frac{1}{2}\times \frac{7}{5}}}$$

Multiply \frac{1}{2} times \frac{7}{5} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{113}{45}}{\frac{5}{4}+\frac{1\times 7}{2\times 5}}}$$

Do the multiplications in the fraction \frac{1\times 7}{2\times 5}.

$$\sqrt{\frac{\frac{113}{45}}{\frac{5}{4}+\frac{7}{10}}}$$

Least common multiple of 4 and 10 is 20. Convert \frac{5}{4} and \frac{7}{10} to fractions with denominator 20.

$$\sqrt{\frac{\frac{113}{45}}{\frac{25}{20}+\frac{14}{20}}}$$

Since \frac{25}{20} and \frac{14}{20} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{113}{45}}{\frac{25+14}{20}}}$$

Add 25 and 14 to get 39.

$$\sqrt{\frac{\frac{113}{45}}{\frac{39}{20}}}$$

Divide \frac{113}{45} by \frac{39}{20} by multiplying \frac{113}{45} by the reciprocal of \frac{39}{20}.

$$\sqrt{\frac{113}{45}\times \frac{20}{39}}$$

Multiply \frac{113}{45} times \frac{20}{39} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{113\times 20}{45\times 39}}$$

Do the multiplications in the fraction \frac{113\times 20}{45\times 39}.

$$\sqrt{\frac{2260}{1755}}$$

Reduce the fraction \frac{2260}{1755} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{452}{351}}$$

Rewrite the square root of the division \sqrt{\frac{452}{351}} as the division of square roots \frac{\sqrt{452}}{\sqrt{351}}.

$$\frac{\sqrt{452}}{\sqrt{351}}$$

Factor 452=2^{2}\times 113. Rewrite the square root of the product \sqrt{2^{2}\times 113} as the product of square roots \sqrt{2^{2}}\sqrt{113}. Take the square root of 2^{2}.

$$\frac{2\sqrt{113}}{\sqrt{351}}$$

Factor 351=3^{2}\times 39. Rewrite the square root of the product \sqrt{3^{2}\times 39} as the product of square roots \sqrt{3^{2}}\sqrt{39}. Take the square root of 3^{2}.

$$\frac{2\sqrt{113}}{3\sqrt{39}}$$

Rationalize the denominator of \frac{2\sqrt{113}}{3\sqrt{39}} by multiplying numerator and denominator by \sqrt{39}.

$$\frac{2\sqrt{113}\sqrt{39}}{3\left(\sqrt{39}\right)^{2}}$$

The square of \sqrt{39} is 39.

$$\frac{2\sqrt{113}\sqrt{39}}{3\times 39}$$

To multiply \sqrt{113} and \sqrt{39}, multiply the numbers under the square root.

$$\frac{2\sqrt{4407}}{3\times 39}$$

Multiply 3 and 39 to get 117.

$$\frac{2\sqrt{4407}}{117}$$