Solve sqrt{3/4*{7/15+[3/5-(3/10*frac{2}{3})^2-(frac{2}{5})^3]:frac{8}{15}-(frac{9}{10})^2}}

$$\sqrt { \frac { 3 } { 4 } \times \{ \frac { 7 } { 15 } + [ \frac { 3 } { 5 } - ( \frac { 3 } { 10 } \times \frac { 2 } { 3 } ) ^ { 2 } - ( \frac { 2 } { 5 } ) ^ { 3 } ] : \frac { 8 } { 15 } - ( \frac { 9 } { 10 } ) ^ { 2 } \} }$$

Evaluate

$$\frac{\sqrt{11}}{5}\approx 0.663324958$$

Solution Steps

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

$$\sqrt{\frac{3}{4}\left(\frac{7}{15}+\frac{\frac{3}{5}-\left(\frac{1}{5}\right)^{2}-\left(\frac{2}{5}\right)^{3}}{\frac{8}{15}}-\left(\frac{9}{10}\right)^{2}\right)}$$

Calculate \frac{1}{5} to the power of 2 and get \frac{1}{25}.

$$\sqrt{\frac{3}{4}\left(\frac{7}{15}+\frac{\frac{3}{5}-\frac{1}{25}-\left(\frac{2}{5}\right)^{3}}{\frac{8}{15}}-\left(\frac{9}{10}\right)^{2}\right)}$$

Subtract \frac{1}{25} from \frac{3}{5} to get \frac{14}{25}.

$$\sqrt{\frac{3}{4}\left(\frac{7}{15}+\frac{\frac{14}{25}-\left(\frac{2}{5}\right)^{3}}{\frac{8}{15}}-\left(\frac{9}{10}\right)^{2}\right)}$$

Calculate \frac{2}{5} to the power of 3 and get \frac{8}{125}.

$$\sqrt{\frac{3}{4}\left(\frac{7}{15}+\frac{\frac{14}{25}-\frac{8}{125}}{\frac{8}{15}}-\left(\frac{9}{10}\right)^{2}\right)}$$

Subtract \frac{8}{125} from \frac{14}{25} to get \frac{62}{125}.

$$\sqrt{\frac{3}{4}\left(\frac{7}{15}+\frac{\frac{62}{125}}{\frac{8}{15}}-\left(\frac{9}{10}\right)^{2}\right)}$$

Divide \frac{62}{125} by \frac{8}{15} by multiplying \frac{62}{125} by the reciprocal of \frac{8}{15}.

$$\sqrt{\frac{3}{4}\left(\frac{7}{15}+\frac{62}{125}\times \frac{15}{8}-\left(\frac{9}{10}\right)^{2}\right)}$$

Multiply \frac{62}{125} and \frac{15}{8} to get \frac{93}{100}.

$$\sqrt{\frac{3}{4}\left(\frac{7}{15}+\frac{93}{100}-\left(\frac{9}{10}\right)^{2}\right)}$$

Add \frac{7}{15} and \frac{93}{100} to get \frac{419}{300}.

$$\sqrt{\frac{3}{4}\left(\frac{419}{300}-\left(\frac{9}{10}\right)^{2}\right)}$$

Calculate \frac{9}{10} to the power of 2 and get \frac{81}{100}.

$$\sqrt{\frac{3}{4}\left(\frac{419}{300}-\frac{81}{100}\right)}$$

Subtract \frac{81}{100} from \frac{419}{300} to get \frac{44}{75}.

$$\sqrt{\frac{3}{4}\times \frac{44}{75}}$$

Multiply \frac{3}{4} and \frac{44}{75} to get \frac{11}{25}.

$$\sqrt{\frac{11}{25}}$$

Rewrite the square root of the division \sqrt{\frac{11}{25}} as the division of square roots \frac{\sqrt{11}}{\sqrt{25}}.

$$\frac{\sqrt{11}}{\sqrt{25}}$$

Calculate the square root of 25 and get 5.

$$\frac{\sqrt{11}}{5}$$