Solve sqrt{[(8/9+1/3)^{2}:(7/3-1/2)^2+(frac{2^3}{15}*frac{25}{2^2}-frac{1}{3})^3]:(frac{2}{9}+frac{8}{3})}

$$\sqrt { [ ( \frac { 8 } { 9 } + \frac { 1 } { 3 } ) ^ { 2 } : ( \frac { 7 } { 3 } - \frac { 1 } { 2 } ) ^ { 2 } + ( \frac { 2 ^ { 3 } } { 15 } \times \frac { 25 } { 2 ^ { 2 } } - \frac { 1 } { 3 } ) ^ { 3 } ] : ( \frac { 2 } { 9 } + \frac { 8 } { 3 } ) }$$

Evaluate

$$\frac{\sqrt{38}}{2}\approx 3.082207001$$

Solution Steps

Add \frac{8}{9} and \frac{1}{3} to get \frac{11}{9}.

$$\sqrt{\frac{\frac{\left(\frac{11}{9}\right)^{2}}{\left(\frac{7}{3}-\frac{1}{2}\right)^{2}}+\left(\frac{2^{3}}{15}\times \frac{25}{2^{2}}-\frac{1}{3}\right)^{3}}{\frac{2}{9}+\frac{8}{3}}}$$

Calculate \frac{11}{9} to the power of 2 and get \frac{121}{81}.

$$\sqrt{\frac{\frac{\frac{121}{81}}{\left(\frac{7}{3}-\frac{1}{2}\right)^{2}}+\left(\frac{2^{3}}{15}\times \frac{25}{2^{2}}-\frac{1}{3}\right)^{3}}{\frac{2}{9}+\frac{8}{3}}}$$

Subtract \frac{1}{2} from \frac{7}{3} to get \frac{11}{6}.

$$\sqrt{\frac{\frac{\frac{121}{81}}{\left(\frac{11}{6}\right)^{2}}+\left(\frac{2^{3}}{15}\times \frac{25}{2^{2}}-\frac{1}{3}\right)^{3}}{\frac{2}{9}+\frac{8}{3}}}$$

Calculate \frac{11}{6} to the power of 2 and get \frac{121}{36}.

$$\sqrt{\frac{\frac{\frac{121}{81}}{\frac{121}{36}}+\left(\frac{2^{3}}{15}\times \frac{25}{2^{2}}-\frac{1}{3}\right)^{3}}{\frac{2}{9}+\frac{8}{3}}}$$

Divide \frac{121}{81} by \frac{121}{36} by multiplying \frac{121}{81} by the reciprocal of \frac{121}{36}.

$$\sqrt{\frac{\frac{121}{81}\times \frac{36}{121}+\left(\frac{2^{3}}{15}\times \frac{25}{2^{2}}-\frac{1}{3}\right)^{3}}{\frac{2}{9}+\frac{8}{3}}}$$

Multiply \frac{121}{81} and \frac{36}{121} to get \frac{4}{9}.

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

Calculate 2 to the power of 3 and get 8.

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

Calculate 2 to the power of 2 and get 4.

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

Multiply \frac{8}{15} and \frac{25}{4} to get \frac{10}{3}.

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

Subtract \frac{1}{3} from \frac{10}{3} to get 3.

$$\sqrt{\frac{\frac{4}{9}+3^{3}}{\frac{2}{9}+\frac{8}{3}}}$$

Calculate 3 to the power of 3 and get 27.

$$\sqrt{\frac{\frac{4}{9}+27}{\frac{2}{9}+\frac{8}{3}}}$$

Add \frac{4}{9} and 27 to get \frac{247}{9}.

$$\sqrt{\frac{\frac{247}{9}}{\frac{2}{9}+\frac{8}{3}}}$$

Add \frac{2}{9} and \frac{8}{3} to get \frac{26}{9}.

$$\sqrt{\frac{\frac{247}{9}}{\frac{26}{9}}}$$

Divide \frac{247}{9} by \frac{26}{9} by multiplying \frac{247}{9} by the reciprocal of \frac{26}{9}.

$$\sqrt{\frac{247}{9}\times \frac{9}{26}}$$

Multiply \frac{247}{9} and \frac{9}{26} to get \frac{19}{2}.

$$\sqrt{\frac{19}{2}}$$

Rewrite the square root of the division \sqrt{\frac{19}{2}} as the division of square roots \frac{\sqrt{19}}{\sqrt{2}}.

$$\frac{\sqrt{19}}{\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{19}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{19}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{19}\sqrt{2}}{2}$$

To multiply \sqrt{19} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{38}}{2}$$