Solve sqrt{1/9*(3/2)+(1/5)*(3+frac{1/3)}{(frac{1}{4})^2+(frac{3}{10})^2*(2+frac{1}{2})^2}*frac{9}{10}+5}

$$\sqrt { \frac { \frac { 1 } { 9 } \cdot ( \frac { 3 } { 2 } ) + ( \frac { 1 } { 5 } ) \cdot ( 3 + \frac { 1 } { 3 } ) } { ( \frac { 1 } { 4 } ) ^ { 2 } + ( \frac { 3 } { 10 } ) ^ { 2 } \cdot ( 2 + \frac { 1 } { 2 } ) ^ { 2 } } \cdot \frac { 9 } { 10 } + 5 }$$

Evaluate

$$\frac{\sqrt{155}}{5}\approx 2.48997992$$

Solution Steps

Multiply \frac{1}{9} and \frac{3}{2} to get \frac{1}{6}.

$$\sqrt{\frac{\frac{1}{6}+\frac{1}{5}\left(3+\frac{1}{3}\right)}{\left(\frac{1}{4}\right)^{2}+\left(\frac{3}{10}\right)^{2}\left(2+\frac{1}{2}\right)^{2}}\times \frac{9}{10}+5}$$

Add 3 and \frac{1}{3} to get \frac{10}{3}.

$$\sqrt{\frac{\frac{1}{6}+\frac{1}{5}\times \frac{10}{3}}{\left(\frac{1}{4}\right)^{2}+\left(\frac{3}{10}\right)^{2}\left(2+\frac{1}{2}\right)^{2}}\times \frac{9}{10}+5}$$

Multiply \frac{1}{5} and \frac{10}{3} to get \frac{2}{3}.

$$\sqrt{\frac{\frac{1}{6}+\frac{2}{3}}{\left(\frac{1}{4}\right)^{2}+\left(\frac{3}{10}\right)^{2}\left(2+\frac{1}{2}\right)^{2}}\times \frac{9}{10}+5}$$

Add \frac{1}{6} and \frac{2}{3} to get \frac{5}{6}.

$$\sqrt{\frac{\frac{5}{6}}{\left(\frac{1}{4}\right)^{2}+\left(\frac{3}{10}\right)^{2}\left(2+\frac{1}{2}\right)^{2}}\times \frac{9}{10}+5}$$

Calculate \frac{1}{4} to the power of 2 and get \frac{1}{16}.

$$\sqrt{\frac{\frac{5}{6}}{\frac{1}{16}+\left(\frac{3}{10}\right)^{2}\left(2+\frac{1}{2}\right)^{2}}\times \frac{9}{10}+5}$$

Calculate \frac{3}{10} to the power of 2 and get \frac{9}{100}.

$$\sqrt{\frac{\frac{5}{6}}{\frac{1}{16}+\frac{9}{100}\left(2+\frac{1}{2}\right)^{2}}\times \frac{9}{10}+5}$$

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

$$\sqrt{\frac{\frac{5}{6}}{\frac{1}{16}+\frac{9}{100}\times \left(\frac{5}{2}\right)^{2}}\times \frac{9}{10}+5}$$

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

$$\sqrt{\frac{\frac{5}{6}}{\frac{1}{16}+\frac{9}{100}\times \frac{25}{4}}\times \frac{9}{10}+5}$$

Multiply \frac{9}{100} and \frac{25}{4} to get \frac{9}{16}.

$$\sqrt{\frac{\frac{5}{6}}{\frac{1}{16}+\frac{9}{16}}\times \frac{9}{10}+5}$$

Add \frac{1}{16} and \frac{9}{16} to get \frac{5}{8}.

$$\sqrt{\frac{\frac{5}{6}}{\frac{5}{8}}\times \frac{9}{10}+5}$$

Divide \frac{5}{6} by \frac{5}{8} by multiplying \frac{5}{6} by the reciprocal of \frac{5}{8}.

$$\sqrt{\frac{5}{6}\times \frac{8}{5}\times \frac{9}{10}+5}$$

Multiply \frac{5}{6} and \frac{8}{5} to get \frac{4}{3}.

$$\sqrt{\frac{4}{3}\times \frac{9}{10}+5}$$

Multiply \frac{4}{3} and \frac{9}{10} to get \frac{6}{5}.

$$\sqrt{\frac{6}{5}+5}$$

Add \frac{6}{5} and 5 to get \frac{31}{5}.

$$\sqrt{\frac{31}{5}}$$

Rewrite the square root of the division \sqrt{\frac{31}{5}} as the division of square roots \frac{\sqrt{31}}{\sqrt{5}}.

$$\frac{\sqrt{31}}{\sqrt{5}}$$

Rationalize the denominator of \frac{\sqrt{31}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\sqrt{31}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{31}\sqrt{5}}{5}$$

To multiply \sqrt{31} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\sqrt{155}}{5}$$