Solve sqrt{[4/13*(7-1/5*75/4)]:[16/3*(frac{4}{3}+frac{5}{6}:frac{1}{2})]+sqrt{(frac{53}{5}-frac{63}{20}-5)*(1+frac{1}{4}}}

$$\sqrt { [ \frac { 4 } { 13 } \times ( 7 - \frac { 1 } { 5 } \times \frac { 75 } { 4 } ) ] : [ \frac { 16 } { 3 } \times ( \frac { 4 } { 3 } + \frac { 5 } { 6 } : \frac { 1 } { 2 } ) ] + \sqrt { ( \frac { 53 } { 5 } - \frac { 63 } { 20 } - 5 ) \times ( 1 + \frac { 1 } { 4 } } }$$

Evaluate

$$\frac{\sqrt{29}}{4}\approx 1.346291202$$

Solution Steps

Multiply \frac{1}{5} times \frac{75}{4} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{4}{13}\left(7-\frac{1\times 75}{5\times 4}\right)}{\frac{16}{3}\left(\frac{4}{3}+\frac{\frac{5}{6}}{\frac{1}{2}}\right)}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Do the multiplications in the fraction \frac{1\times 75}{5\times 4}.

$$\sqrt{\frac{\frac{4}{13}\left(7-\frac{75}{20}\right)}{\frac{16}{3}\left(\frac{4}{3}+\frac{\frac{5}{6}}{\frac{1}{2}}\right)}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Reduce the fraction \frac{75}{20} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{\frac{4}{13}\left(7-\frac{15}{4}\right)}{\frac{16}{3}\left(\frac{4}{3}+\frac{\frac{5}{6}}{\frac{1}{2}}\right)}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Convert 7 to fraction \frac{28}{4}.

$$\sqrt{\frac{\frac{4}{13}\left(\frac{28}{4}-\frac{15}{4}\right)}{\frac{16}{3}\left(\frac{4}{3}+\frac{\frac{5}{6}}{\frac{1}{2}}\right)}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Since \frac{28}{4} and \frac{15}{4} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{4}{13}\times \frac{28-15}{4}}{\frac{16}{3}\left(\frac{4}{3}+\frac{\frac{5}{6}}{\frac{1}{2}}\right)}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Subtract 15 from 28 to get 13.

$$\sqrt{\frac{\frac{4}{13}\times \frac{13}{4}}{\frac{16}{3}\left(\frac{4}{3}+\frac{\frac{5}{6}}{\frac{1}{2}}\right)}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Cancel out \frac{4}{13} and its reciprocal \frac{13}{4}.

$$\sqrt{\frac{1}{\frac{16}{3}\left(\frac{4}{3}+\frac{\frac{5}{6}}{\frac{1}{2}}\right)}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Divide \frac{5}{6} by \frac{1}{2} by multiplying \frac{5}{6} by the reciprocal of \frac{1}{2}.

$$\sqrt{\frac{1}{\frac{16}{3}\left(\frac{4}{3}+\frac{5}{6}\times 2\right)}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Express \frac{5}{6}\times 2 as a single fraction.

$$\sqrt{\frac{1}{\frac{16}{3}\left(\frac{4}{3}+\frac{5\times 2}{6}\right)}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Multiply 5 and 2 to get 10.

$$\sqrt{\frac{1}{\frac{16}{3}\left(\frac{4}{3}+\frac{10}{6}\right)}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Reduce the fraction \frac{10}{6} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{1}{\frac{16}{3}\left(\frac{4}{3}+\frac{5}{3}\right)}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Since \frac{4}{3} and \frac{5}{3} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1}{\frac{16}{3}\times \frac{4+5}{3}}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Add 4 and 5 to get 9.

$$\sqrt{\frac{1}{\frac{16}{3}\times \frac{9}{3}}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Divide 9 by 3 to get 3.

$$\sqrt{\frac{1}{\frac{16}{3}\times 3}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Cancel out 3 and 3.

$$\sqrt{\frac{1}{16}+\sqrt{\left(\frac{53}{5}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Least common multiple of 5 and 20 is 20. Convert \frac{53}{5} and \frac{63}{20} to fractions with denominator 20.

$$\sqrt{\frac{1}{16}+\sqrt{\left(\frac{212}{20}-\frac{63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Since \frac{212}{20} and \frac{63}{20} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{1}{16}+\sqrt{\left(\frac{212-63}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Subtract 63 from 212 to get 149.

$$\sqrt{\frac{1}{16}+\sqrt{\left(\frac{149}{20}-5\right)\left(1+\frac{1}{4}\right)}}$$

Convert 5 to fraction \frac{100}{20}.

$$\sqrt{\frac{1}{16}+\sqrt{\left(\frac{149}{20}-\frac{100}{20}\right)\left(1+\frac{1}{4}\right)}}$$

Since \frac{149}{20} and \frac{100}{20} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{1}{16}+\sqrt{\frac{149-100}{20}\left(1+\frac{1}{4}\right)}}$$

Subtract 100 from 149 to get 49.

$$\sqrt{\frac{1}{16}+\sqrt{\frac{49}{20}\left(1+\frac{1}{4}\right)}}$$

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

$$\sqrt{\frac{1}{16}+\sqrt{\frac{49}{20}\left(\frac{4}{4}+\frac{1}{4}\right)}}$$

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

$$\sqrt{\frac{1}{16}+\sqrt{\frac{49}{20}\times \frac{4+1}{4}}}$$

Add 4 and 1 to get 5.

$$\sqrt{\frac{1}{16}+\sqrt{\frac{49}{20}\times \frac{5}{4}}}$$

Multiply \frac{49}{20} times \frac{5}{4} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{1}{16}+\sqrt{\frac{49\times 5}{20\times 4}}}$$

Do the multiplications in the fraction \frac{49\times 5}{20\times 4}.

$$\sqrt{\frac{1}{16}+\sqrt{\frac{245}{80}}}$$

Reduce the fraction \frac{245}{80} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{1}{16}+\sqrt{\frac{49}{16}}}$$

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

$$\sqrt{\frac{1}{16}+\frac{7}{4}}$$

Least common multiple of 16 and 4 is 16. Convert \frac{1}{16} and \frac{7}{4} to fractions with denominator 16.

$$\sqrt{\frac{1}{16}+\frac{28}{16}}$$

Since \frac{1}{16} and \frac{28}{16} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1+28}{16}}$$

Add 1 and 28 to get 29.

$$\sqrt{\frac{29}{16}}$$

Rewrite the square root of the division \sqrt{\frac{29}{16}} as the division of square roots \frac{\sqrt{29}}{\sqrt{16}}.

$$\frac{\sqrt{29}}{\sqrt{16}}$$

Calculate the square root of 16 and get 4.

$$\frac{\sqrt{29}}{4}$$