Solve [(-4/3)^{-1}*(4/3)^2+7/9]:[(-3)^-1:(-3)^-2+1/2]-3^-2

$$[ ( - \frac { 4 } { 3 } ) ^ { - 1 } \cdot ( \frac { 4 } { 3 } ) ^ { 2 } + \frac { 7 } { 9 } ] : [ ( - 3 ) ^ { - 1 } : ( - 3 ) ^ { - 2 } + \frac { 1 } { 2 } ] - 3 ^ { - 2 }$$

Evaluate

$$\frac{1}{9}\approx 0.111111111$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -2 from -1 to get 1.

$$\frac{\left(-\frac{4}{3}\right)^{-1}\times \left(\frac{4}{3}\right)^{2}+\frac{7}{9}}{\left(-3\right)^{1}+\frac{1}{2}}-3^{-2}$$

Calculate -\frac{4}{3} to the power of -1 and get -\frac{3}{4}.

$$\frac{-\frac{3}{4}\times \left(\frac{4}{3}\right)^{2}+\frac{7}{9}}{\left(-3\right)^{1}+\frac{1}{2}}-3^{-2}$$

Calculate \frac{4}{3} to the power of 2 and get \frac{16}{9}.

$$\frac{-\frac{3}{4}\times \frac{16}{9}+\frac{7}{9}}{\left(-3\right)^{1}+\frac{1}{2}}-3^{-2}$$

Multiply -\frac{3}{4} and \frac{16}{9} to get -\frac{4}{3}.

$$\frac{-\frac{4}{3}+\frac{7}{9}}{\left(-3\right)^{1}+\frac{1}{2}}-3^{-2}$$

Add -\frac{4}{3} and \frac{7}{9} to get -\frac{5}{9}.

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

Calculate -3 to the power of 1 and get -3.

$$\frac{-\frac{5}{9}}{-3+\frac{1}{2}}-3^{-2}$$

Add -3 and \frac{1}{2} to get -\frac{5}{2}.

$$\frac{-\frac{5}{9}}{-\frac{5}{2}}-3^{-2}$$

Divide -\frac{5}{9} by -\frac{5}{2} by multiplying -\frac{5}{9} by the reciprocal of -\frac{5}{2}.

$$-\frac{5}{9}\left(-\frac{2}{5}\right)-3^{-2}$$

Multiply -\frac{5}{9} and -\frac{2}{5} to get \frac{2}{9}.

$$\frac{2}{9}-3^{-2}$$

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

$$\frac{2}{9}-\frac{1}{9}$$

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

$$\frac{1}{9}$$

Factor

$$\frac{1}{3 ^ {2}} = 0.1111111111111111$$