Solve (-1/3)^3*(-2z/3)^-2div{2^-3*[2-(2/3)^-2]}

$$( - \frac { 1 } { 3 } ) ^ { 3 } \times ( - 2 \frac { z } { 3 } ) ^ { - 2 } \div \{ 2 ^ { - 3 } \times [ 2 - ( \frac { 2 } { 3 } ) ^ { - 2 } ] \}$$

Evaluate

$$\frac{8}{3z^{2}}$$

Solution Steps

Calculate -\frac{1}{3} to the power of 3 and get -\frac{1}{27}.

$$\frac{-\frac{1}{27}\left(-2\times \frac{z}{3}\right)^{-2}}{2^{-3}\left(2-\left(\frac{2}{3}\right)^{-2}\right)}$$

Express -2\times \frac{z}{3} as a single fraction.

$$\frac{-\frac{1}{27}\times \left(\frac{-2z}{3}\right)^{-2}}{2^{-3}\left(2-\left(\frac{2}{3}\right)^{-2}\right)}$$

To raise \frac{-2z}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{-\frac{1}{27}\times \frac{\left(-2z\right)^{-2}}{3^{-2}}}{2^{-3}\left(2-\left(\frac{2}{3}\right)^{-2}\right)}$$

Multiply -\frac{1}{27} times \frac{\left(-2z\right)^{-2}}{3^{-2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{-\left(-2z\right)^{-2}}{27\times 3^{-2}}}{2^{-3}\left(2-\left(\frac{2}{3}\right)^{-2}\right)}$$

Calculate 2 to the power of -3 and get \frac{1}{8}.

$$\frac{\frac{-\left(-2z\right)^{-2}}{27\times 3^{-2}}}{\frac{1}{8}\left(2-\left(\frac{2}{3}\right)^{-2}\right)}$$

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

$$\frac{\frac{-\left(-2z\right)^{-2}}{27\times 3^{-2}}}{\frac{1}{8}\left(2-\frac{9}{4}\right)}$$

Subtract \frac{9}{4} from 2 to get -\frac{1}{4}.

$$\frac{\frac{-\left(-2z\right)^{-2}}{27\times 3^{-2}}}{\frac{1}{8}\left(-\frac{1}{4}\right)}$$

Multiply \frac{1}{8} and -\frac{1}{4} to get -\frac{1}{32}.

$$\frac{\frac{-\left(-2z\right)^{-2}}{27\times 3^{-2}}}{-\frac{1}{32}}$$

Divide \frac{-\left(-2z\right)^{-2}}{27\times 3^{-2}} by -\frac{1}{32} by multiplying \frac{-\left(-2z\right)^{-2}}{27\times 3^{-2}} by the reciprocal of -\frac{1}{32}.

$$\frac{-\left(-2z\right)^{-2}\times 32}{27\times 3^{-2}\left(-1\right)}$$

Cancel out -1 in both numerator and denominator.

$$\frac{32\left(-2z\right)^{-2}}{27\times 3^{-2}}$$

Expand \left(-2z\right)^{-2}.

$$\frac{32\left(-2\right)^{-2}z^{-2}}{27\times 3^{-2}}$$

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

$$\frac{32\times \frac{1}{4}z^{-2}}{27\times 3^{-2}}$$

Multiply 32 and \frac{1}{4} to get 8.

$$\frac{8z^{-2}}{27\times 3^{-2}}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{8z^{-2}}{27\times \frac{1}{9}}$$

Multiply 27 and \frac{1}{9} to get 3.

$$\frac{8z^{-2}}{3}$$

Expand

$$\frac{8}{3z^{2}}$$

Solution Steps

Calculate -\frac{1}{3} to the power of 3 and get -\frac{1}{27}.

$$\frac{-\frac{1}{27}\left(-2\times \frac{z}{3}\right)^{-2}}{2^{-3}\left(2-\left(\frac{2}{3}\right)^{-2}\right)}$$

Express -2\times \frac{z}{3} as a single fraction.

$$\frac{-\frac{1}{27}\times \left(\frac{-2z}{3}\right)^{-2}}{2^{-3}\left(2-\left(\frac{2}{3}\right)^{-2}\right)}$$

To raise \frac{-2z}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{-\frac{1}{27}\times \frac{\left(-2z\right)^{-2}}{3^{-2}}}{2^{-3}\left(2-\left(\frac{2}{3}\right)^{-2}\right)}$$

Multiply -\frac{1}{27} times \frac{\left(-2z\right)^{-2}}{3^{-2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{-\left(-2z\right)^{-2}}{27\times 3^{-2}}}{2^{-3}\left(2-\left(\frac{2}{3}\right)^{-2}\right)}$$

Calculate 2 to the power of -3 and get \frac{1}{8}.

$$\frac{\frac{-\left(-2z\right)^{-2}}{27\times 3^{-2}}}{\frac{1}{8}\left(2-\left(\frac{2}{3}\right)^{-2}\right)}$$

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

$$\frac{\frac{-\left(-2z\right)^{-2}}{27\times 3^{-2}}}{\frac{1}{8}\left(2-\frac{9}{4}\right)}$$

Subtract \frac{9}{4} from 2 to get -\frac{1}{4}.

$$\frac{\frac{-\left(-2z\right)^{-2}}{27\times 3^{-2}}}{\frac{1}{8}\left(-\frac{1}{4}\right)}$$

Multiply \frac{1}{8} and -\frac{1}{4} to get -\frac{1}{32}.

$$\frac{\frac{-\left(-2z\right)^{-2}}{27\times 3^{-2}}}{-\frac{1}{32}}$$

Divide \frac{-\left(-2z\right)^{-2}}{27\times 3^{-2}} by -\frac{1}{32} by multiplying \frac{-\left(-2z\right)^{-2}}{27\times 3^{-2}} by the reciprocal of -\frac{1}{32}.

$$\frac{-\left(-2z\right)^{-2}\times 32}{27\times 3^{-2}\left(-1\right)}$$

Cancel out -1 in both numerator and denominator.

$$\frac{32\left(-2z\right)^{-2}}{27\times 3^{-2}}$$

Expand \left(-2z\right)^{-2}.

$$\frac{32\left(-2\right)^{-2}z^{-2}}{27\times 3^{-2}}$$

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

$$\frac{32\times \frac{1}{4}z^{-2}}{27\times 3^{-2}}$$

Multiply 32 and \frac{1}{4} to get 8.

$$\frac{8z^{-2}}{27\times 3^{-2}}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{8z^{-2}}{27\times \frac{1}{9}}$$

Multiply 27 and \frac{1}{9} to get 3.

$$\frac{8z^{-2}}{3}$$