Solve (2)^{-1}+[2/3-2^-1*((2/3)^-2*2/3+5^0)]^-1=

$$( 2 ) ^ { - 1 } + [ \frac { 2 } { 3 } - 2 ^ { - 1 } \cdot ( ( \frac { 2 } { 3 } ) ^ { - 2 } \cdot \frac { 2 } { 3 } + 5 ^ { 0 } ) ] ^ { - 1 } =$$

Evaluate

$$-\frac{17}{14}\approx -1.214285714$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -2 and 1 to get -1.

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

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

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

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

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

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

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

Calculate 5 to the power of 0 and get 1.

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

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

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

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

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

Subtract \frac{5}{4} from \frac{2}{3} to get -\frac{7}{12}.

$$\frac{1}{2}+\left(-\frac{7}{12}\right)^{-1}$$

Calculate -\frac{7}{12} to the power of -1 and get -\frac{12}{7}.

$$\frac{1}{2}-\frac{12}{7}$$

Subtract \frac{12}{7} from \frac{1}{2} to get -\frac{17}{14}.

$$-\frac{17}{14}$$

Factor

$$-\frac{17}{14} = -1\frac{3}{14} = -1.2142857142857142$$