Solve sqrt{[1/20*(5/6+5/12)^2+(9/2-frac{15}{4})^3]:{[frac{10}{3}-frac{4}{3}-(frac{16}{5}-frac{7}{10}-frac{3}{4})]*frac{27}{2}}}

$$\sqrt { [ \frac { 1 } { 20 } \cdot ( \frac { 5 } { 6 } + \frac { 5 } { 12 } ) ^ { 2 } + ( \frac { 9 } { 2 } - \frac { 15 } { 4 } ) ^ { 3 } ] : \{ [ \frac { 10 } { 3 } - \frac { 4 } { 3 } - ( \frac { 16 } { 5 } - \frac { 7 } { 10 } - \frac { 3 } { 4 } ) ] \cdot \frac { 27 } { 2 } \} }$$

Evaluate

$$\frac{2\sqrt{3}}{9}\approx 0.384900179$$

Solution Steps

Add \frac{5}{6} and \frac{5}{12} to get \frac{5}{4}.

$$\sqrt{\frac{\frac{1}{20}\times \left(\frac{5}{4}\right)^{2}+\left(\frac{9}{2}-\frac{15}{4}\right)^{3}}{\left(\frac{10}{3}-\frac{4}{3}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \frac{27}{2}}}$$

Calculate \frac{5}{4} to the power of 2 and get \frac{25}{16}.

$$\sqrt{\frac{\frac{1}{20}\times \frac{25}{16}+\left(\frac{9}{2}-\frac{15}{4}\right)^{3}}{\left(\frac{10}{3}-\frac{4}{3}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \frac{27}{2}}}$$

Multiply \frac{1}{20} and \frac{25}{16} to get \frac{5}{64}.

$$\sqrt{\frac{\frac{5}{64}+\left(\frac{9}{2}-\frac{15}{4}\right)^{3}}{\left(\frac{10}{3}-\frac{4}{3}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \frac{27}{2}}}$$

Subtract \frac{15}{4} from \frac{9}{2} to get \frac{3}{4}.

$$\sqrt{\frac{\frac{5}{64}+\left(\frac{3}{4}\right)^{3}}{\left(\frac{10}{3}-\frac{4}{3}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \frac{27}{2}}}$$

Calculate \frac{3}{4} to the power of 3 and get \frac{27}{64}.

$$\sqrt{\frac{\frac{5}{64}+\frac{27}{64}}{\left(\frac{10}{3}-\frac{4}{3}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \frac{27}{2}}}$$

Add \frac{5}{64} and \frac{27}{64} to get \frac{1}{2}.

$$\sqrt{\frac{\frac{1}{2}}{\left(\frac{10}{3}-\frac{4}{3}-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \frac{27}{2}}}$$

Subtract \frac{4}{3} from \frac{10}{3} to get 2.

$$\sqrt{\frac{\frac{1}{2}}{\left(2-\left(\frac{16}{5}-\frac{7}{10}-\frac{3}{4}\right)\right)\times \frac{27}{2}}}$$

Subtract \frac{7}{10} from \frac{16}{5} to get \frac{5}{2}.

$$\sqrt{\frac{\frac{1}{2}}{\left(2-\left(\frac{5}{2}-\frac{3}{4}\right)\right)\times \frac{27}{2}}}$$

Subtract \frac{3}{4} from \frac{5}{2} to get \frac{7}{4}.

$$\sqrt{\frac{\frac{1}{2}}{\left(2-\frac{7}{4}\right)\times \frac{27}{2}}}$$

Subtract \frac{7}{4} from 2 to get \frac{1}{4}.

$$\sqrt{\frac{\frac{1}{2}}{\frac{1}{4}\times \frac{27}{2}}}$$

Multiply \frac{1}{4} and \frac{27}{2} to get \frac{27}{8}.

$$\sqrt{\frac{\frac{1}{2}}{\frac{27}{8}}}$$

Divide \frac{1}{2} by \frac{27}{8} by multiplying \frac{1}{2} by the reciprocal of \frac{27}{8}.

$$\sqrt{\frac{1}{2}\times \frac{8}{27}}$$

Multiply \frac{1}{2} and \frac{8}{27} to get \frac{4}{27}.

$$\sqrt{\frac{4}{27}}$$

Rewrite the square root of the division \sqrt{\frac{4}{27}} as the division of square roots \frac{\sqrt{4}}{\sqrt{27}}.

$$\frac{\sqrt{4}}{\sqrt{27}}$$

Calculate the square root of 4 and get 2.

$$\frac{2}{\sqrt{27}}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 3 and 3 to get 9.

$$\frac{2\sqrt{3}}{9}$$