Solve sqrt{7/8-5/6}+5/9-7/16+frac{5}{12}-frac{1}{72}

$$\sqrt{ \frac{ 7 }{ 8 } - \frac{ 5 }{ 6 } } + \frac{ 5 }{ 9 } - \frac{ 7 }{ 16 } + \frac{ 5 }{ 12 } - \frac{ 1 }{ 72 }$$

Evaluate

$$\frac{\sqrt{6}}{12}+\frac{25}{48}\approx 0.724957479$$

Short Solution Steps

Least common multiple of 8 and 6 is 24. Convert \frac{7}{8} and \frac{5}{6} to fractions with denominator 24.

$$\sqrt{\frac{21}{24}-\frac{20}{24}}+\frac{5}{9}-\frac{7}{16}+\frac{5}{12}-\frac{1}{72}$$

Since \frac{21}{24} and \frac{20}{24} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{21-20}{24}}+\frac{5}{9}-\frac{7}{16}+\frac{5}{12}-\frac{1}{72}$$

Subtract 20 from 21 to get 1.

$$\sqrt{\frac{1}{24}}+\frac{5}{9}-\frac{7}{16}+\frac{5}{12}-\frac{1}{72}$$

Rewrite the square root of the division \sqrt{\frac{1}{24}} as the division of square roots \frac{\sqrt{1}}{\sqrt{24}}.

$$\frac{\sqrt{1}}{\sqrt{24}}+\frac{5}{9}-\frac{7}{16}+\frac{5}{12}-\frac{1}{72}$$

Calculate the square root of 1 and get 1.

$$\frac{1}{\sqrt{24}}+\frac{5}{9}-\frac{7}{16}+\frac{5}{12}-\frac{1}{72}$$

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}.

$$\frac{1}{2\sqrt{6}}+\frac{5}{9}-\frac{7}{16}+\frac{5}{12}-\frac{1}{72}$$

Rationalize the denominator of \frac{1}{2\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\sqrt{6}}{2\left(\sqrt{6}\right)^{2}}+\frac{5}{9}-\frac{7}{16}+\frac{5}{12}-\frac{1}{72}$$

The square of \sqrt{6} is 6.

$$\frac{\sqrt{6}}{2\times 6}+\frac{5}{9}-\frac{7}{16}+\frac{5}{12}-\frac{1}{72}$$

Multiply 2 and 6 to get 12.

$$\frac{\sqrt{6}}{12}+\frac{5}{9}-\frac{7}{16}+\frac{5}{12}-\frac{1}{72}$$

Least common multiple of 9 and 16 is 144. Convert \frac{5}{9} and \frac{7}{16} to fractions with denominator 144.

$$\frac{\sqrt{6}}{12}+\frac{80}{144}-\frac{63}{144}+\frac{5}{12}-\frac{1}{72}$$

Since \frac{80}{144} and \frac{63}{144} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{6}}{12}+\frac{80-63}{144}+\frac{5}{12}-\frac{1}{72}$$

Subtract 63 from 80 to get 17.

$$\frac{\sqrt{6}}{12}+\frac{17}{144}+\frac{5}{12}-\frac{1}{72}$$

Least common multiple of 144 and 12 is 144. Convert \frac{17}{144} and \frac{5}{12} to fractions with denominator 144.

$$\frac{\sqrt{6}}{12}+\frac{17}{144}+\frac{60}{144}-\frac{1}{72}$$

Since \frac{17}{144} and \frac{60}{144} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{6}}{12}+\frac{17+60}{144}-\frac{1}{72}$$

Add 17 and 60 to get 77.

$$\frac{\sqrt{6}}{12}+\frac{77}{144}-\frac{1}{72}$$

Least common multiple of 144 and 72 is 144. Convert \frac{77}{144} and \frac{1}{72} to fractions with denominator 144.

$$\frac{\sqrt{6}}{12}+\frac{77}{144}-\frac{2}{144}$$

Since \frac{77}{144} and \frac{2}{144} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{6}}{12}+\frac{77-2}{144}$$

Subtract 2 from 77 to get 75.

$$\frac{\sqrt{6}}{12}+\frac{75}{144}$$

Reduce the fraction \frac{75}{144} to lowest terms by extracting and canceling out 3.

$$\frac{\sqrt{6}}{12}+\frac{25}{48}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 12 and 48 is 48. Multiply \frac{\sqrt{6}}{12} times \frac{4}{4}.

$$\frac{4\sqrt{6}}{48}+\frac{25}{48}$$

Since \frac{4\sqrt{6}}{48} and \frac{25}{48} have the same denominator, add them by adding their numerators.

$$\frac{4\sqrt{6}+25}{48}$$

Factor

$$\frac{4 \sqrt{6} + 25}{48} = 0.7249574785652649$$