Solve [(-1)^{2}-2^-3div1/2]^-1-frac{sqrt{3}}{3}(2sqrt{3}-3sqrt{3})-3^0*3^-2

$$[ ( - 1 ) ^ { 2 } - 2 ^ { - 3 } \div \frac { 1 } { 2 } ] ^ { - 1 } - \frac { \sqrt { 3 } } { 3 } ( 2 \sqrt { 3 } - 3 \sqrt { 3 } ) - 3 ^ { 0 } \cdot 3 ^ { - 2 }$$

Evaluate

$$\frac{20}{9}\approx 2.222222222$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 0 and -2 to get -2.

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

Calculate -1 to the power of 2 and get 1.

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

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

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

Divide \frac{1}{8} by \frac{1}{2} by multiplying \frac{1}{8} by the reciprocal of \frac{1}{2}.

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

Multiply \frac{1}{8} and 2 to get \frac{1}{4}.

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

Subtract \frac{1}{4} from 1 to get \frac{3}{4}.

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

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

$$\frac{4}{3}-\frac{\sqrt{3}}{3}\left(2\sqrt{3}-3\sqrt{3}\right)-3^{-2}$$

Combine 2\sqrt{3} and -3\sqrt{3} to get -\sqrt{3}.

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

Express \frac{\sqrt{3}}{3}\sqrt{3} as a single fraction.

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

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

$$\frac{4}{3}-\frac{\sqrt{3}\sqrt{3}}{3}\left(-1\right)-\frac{1}{9}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{4}{3}-\frac{3}{3}\left(-1\right)-\frac{1}{9}$$

Divide 3 by 3 to get 1.

$$\frac{4}{3}-1\left(-1\right)-\frac{1}{9}$$

Multiply 1 and -1 to get -1.

$$\frac{4}{3}-\left(-1\right)-\frac{1}{9}$$

The opposite of -1 is 1.

$$\frac{4}{3}+1-\frac{1}{9}$$

Add \frac{4}{3} and 1 to get \frac{7}{3}.

$$\frac{7}{3}-\frac{1}{9}$$

Subtract \frac{1}{9} from \frac{7}{3} to get \frac{20}{9}.

$$\frac{20}{9}$$

Factor

$$\frac{2 ^ {2} \cdot 5}{3 ^ {2}} = 2\frac{2}{9} = 2.2222222222222223$$