Solve sqrt{{[(2/3+1/2)-1/6]^2}^3+1-(2/3+frac{7}{6}+frac{2}{12})+[15*(frac{1}{3}+frac{1}{5})]^2}

$$\sqrt { \{ [ ( \frac { 2 } { 3 } + \frac { 1 } { 2 } ) - \frac { 1 } { 6 } ] ^ { 2 } \} ^ { 3 } + 1 - ( \frac { 2 } { 3 } + \frac { 7 } { 6 } + \frac { 2 } { 12 } ) + [ 15 \times ( \frac { 1 } { 3 } + \frac { 1 } { 5 } ) ] ^ { 2 } }$$

Evaluate

$$8$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\sqrt{\left(\frac{2}{3}+\frac{1}{2}-\frac{1}{6}\right)^{6}+1-\left(\frac{2}{3}+\frac{7}{6}+\frac{2}{12}\right)+\left(15\left(\frac{1}{3}+\frac{1}{5}\right)\right)^{2}}$$

Add \frac{2}{3} and \frac{1}{2} to get \frac{7}{6}.

$$\sqrt{\left(\frac{7}{6}-\frac{1}{6}\right)^{6}+1-\left(\frac{2}{3}+\frac{7}{6}+\frac{2}{12}\right)+\left(15\left(\frac{1}{3}+\frac{1}{5}\right)\right)^{2}}$$

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

$$\sqrt{1^{6}+1-\left(\frac{2}{3}+\frac{7}{6}+\frac{2}{12}\right)+\left(15\left(\frac{1}{3}+\frac{1}{5}\right)\right)^{2}}$$

Calculate 1 to the power of 6 and get 1.

$$\sqrt{1+1-\left(\frac{2}{3}+\frac{7}{6}+\frac{2}{12}\right)+\left(15\left(\frac{1}{3}+\frac{1}{5}\right)\right)^{2}}$$

Add 1 and 1 to get 2.

$$\sqrt{2-\left(\frac{2}{3}+\frac{7}{6}+\frac{2}{12}\right)+\left(15\left(\frac{1}{3}+\frac{1}{5}\right)\right)^{2}}$$

Add \frac{2}{3} and \frac{7}{6} to get \frac{11}{6}.

$$\sqrt{2-\left(\frac{11}{6}+\frac{2}{12}\right)+\left(15\left(\frac{1}{3}+\frac{1}{5}\right)\right)^{2}}$$

Reduce the fraction \frac{2}{12} to lowest terms by extracting and canceling out 2.

$$\sqrt{2-\left(\frac{11}{6}+\frac{1}{6}\right)+\left(15\left(\frac{1}{3}+\frac{1}{5}\right)\right)^{2}}$$

Add \frac{11}{6} and \frac{1}{6} to get 2.

$$\sqrt{2-2+\left(15\left(\frac{1}{3}+\frac{1}{5}\right)\right)^{2}}$$

Subtract 2 from 2 to get 0.

$$\sqrt{0+\left(15\left(\frac{1}{3}+\frac{1}{5}\right)\right)^{2}}$$

Add \frac{1}{3} and \frac{1}{5} to get \frac{8}{15}.

$$\sqrt{0+\left(15\times \frac{8}{15}\right)^{2}}$$

Multiply 15 and \frac{8}{15} to get 8.

$$\sqrt{0+8^{2}}$$

Calculate 8 to the power of 2 and get 64.

$$\sqrt{0+64}$$

Add 0 and 64 to get 64.

$$\sqrt{64}$$

Calculate the square root of 64 and get 8.

$$8$$

Factor

$$2^{3}$$