Solve {-[-(1/2+1)^{2}]-5/4}^3:(-1/2)^2+(1/2+frac{5}{4})^2:(frac{1}{8}-1)^2

$$\{ - [ - ( \frac { 1 } { 2 } + 1 ) ^ { 2 } ] - \frac { 5 } { 4 } \} ^ { 3 } : ( - \frac { 1 } { 2 } ) ^ { 2 } + ( \frac { 1 } { 2 } + \frac { 5 } { 4 } ) ^ { 2 } : ( \frac { 1 } { 8 } - 1 ) ^ { 2 }$$

Evaluate

$$8$$

Solution Steps

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

$$\frac{\left(-\left(-\left(\frac{3}{2}\right)^{2}\right)-\frac{5}{4}\right)^{3}}{\left(-\frac{1}{2}\right)^{2}}+\frac{\left(\frac{1}{2}+\frac{5}{4}\right)^{2}}{\left(\frac{1}{8}-1\right)^{2}}$$

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

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

The opposite of -\frac{9}{4} is \frac{9}{4}.

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

Subtract \frac{5}{4} from \frac{9}{4} to get 1.

$$\frac{1^{3}}{\left(-\frac{1}{2}\right)^{2}}+\frac{\left(\frac{1}{2}+\frac{5}{4}\right)^{2}}{\left(\frac{1}{8}-1\right)^{2}}$$

Calculate 1 to the power of 3 and get 1.

$$\frac{1}{\left(-\frac{1}{2}\right)^{2}}+\frac{\left(\frac{1}{2}+\frac{5}{4}\right)^{2}}{\left(\frac{1}{8}-1\right)^{2}}$$

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

$$\frac{1}{\frac{1}{4}}+\frac{\left(\frac{1}{2}+\frac{5}{4}\right)^{2}}{\left(\frac{1}{8}-1\right)^{2}}$$

Divide 1 by \frac{1}{4} by multiplying 1 by the reciprocal of \frac{1}{4}.

$$1\times 4+\frac{\left(\frac{1}{2}+\frac{5}{4}\right)^{2}}{\left(\frac{1}{8}-1\right)^{2}}$$

Multiply 1 and 4 to get 4.

$$4+\frac{\left(\frac{1}{2}+\frac{5}{4}\right)^{2}}{\left(\frac{1}{8}-1\right)^{2}}$$

Add \frac{1}{2} and \frac{5}{4} to get \frac{7}{4}.

$$4+\frac{\left(\frac{7}{4}\right)^{2}}{\left(\frac{1}{8}-1\right)^{2}}$$

Calculate \frac{7}{4} to the power of 2 and get \frac{49}{16}.

$$4+\frac{\frac{49}{16}}{\left(\frac{1}{8}-1\right)^{2}}$$

Subtract 1 from \frac{1}{8} to get -\frac{7}{8}.

$$4+\frac{\frac{49}{16}}{\left(-\frac{7}{8}\right)^{2}}$$

Calculate -\frac{7}{8} to the power of 2 and get \frac{49}{64}.

$$4+\frac{\frac{49}{16}}{\frac{49}{64}}$$

Divide \frac{49}{16} by \frac{49}{64} by multiplying \frac{49}{16} by the reciprocal of \frac{49}{64}.

$$4+\frac{49}{16}\times \frac{64}{49}$$

Multiply \frac{49}{16} and \frac{64}{49} to get 4.

$$4+4$$

Add 4 and 4 to get 8.

$$8$$

Factor

$$2^{3}$$