Solve [(-2/3)^{3}]^{2}:[(-2/3)^{11}:(-2/3)^{5}]+{[(-1/2)^{6}(-frac{1}{2})^5]:[(-frac{1}{2})^4]^2}^-1

$$[ ( - \frac { 2 } { 3 } ) ^ { 3 } ] ^ { 2 } : [ ( - \frac { 2 } { 3 } ) ^ { 11 } : ( - \frac { 2 } { 3 } ) ^ { 5 } ] + \{ [ ( - \frac { 1 } { 2 } ) ^ { 6 } ( - \frac { 1 } { 2 } ) ^ { 5 } ] : [ ( - \frac { 1 } { 2 } ) ^ { 4 } ] ^ { 2 } \} ^ { - 1 }$$

Evaluate

$$-7$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{\left(-\frac{2}{3}\right)^{6}}{\frac{\left(-\frac{2}{3}\right)^{11}}{\left(-\frac{2}{3}\right)^{5}}}+\left(\frac{\left(-\frac{1}{2}\right)^{6}\left(-\frac{1}{2}\right)^{5}}{\left(\left(-\frac{1}{2}\right)^{4}\right)^{2}}\right)^{-1}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 5 from 11 to get 6.

$$\frac{\left(-\frac{2}{3}\right)^{6}}{\left(-\frac{2}{3}\right)^{6}}+\left(\frac{\left(-\frac{1}{2}\right)^{6}\left(-\frac{1}{2}\right)^{5}}{\left(\left(-\frac{1}{2}\right)^{4}\right)^{2}}\right)^{-1}$$

Divide \left(-\frac{2}{3}\right)^{6} by \left(-\frac{2}{3}\right)^{6} to get 1.

$$1+\left(\frac{\left(-\frac{1}{2}\right)^{6}\left(-\frac{1}{2}\right)^{5}}{\left(\left(-\frac{1}{2}\right)^{4}\right)^{2}}\right)^{-1}$$

To multiply powers of the same base, add their exponents. Add 6 and 5 to get 11.

$$1+\left(\frac{\left(-\frac{1}{2}\right)^{11}}{\left(\left(-\frac{1}{2}\right)^{4}\right)^{2}}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$1+\left(\frac{\left(-\frac{1}{2}\right)^{11}}{\left(-\frac{1}{2}\right)^{8}}\right)^{-1}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 8 from 11 to get 3.

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

To raise a power to another power, multiply the exponents. Multiply 3 and -1 to get -3.

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

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

$$1-8$$

Subtract 8 from 1 to get -7.

$$-7$$

Factor

$$-7$$