91Ó°ÊÓ

First, look at the expression \( \left(-\frac{2}{3}\right)^{3} \). This means we need to multiply \(-\frac{2}{3}\) by itself three times. Thus, we have: \[\left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right).\]Let's compute this step-by-step: 1. Multiply the first two fractions: \[\left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) = \frac{4}{9}.\] This is because the negative signs cancel each other (\(- \times - = +\)) and multiplying fractions is done by multiplying the numerators and the denominators. 2. Now, multiply the resulting fraction by the third \(-\frac{2}{3}\): \[\frac{4}{9} \times \left(-\frac{2}{3}\right) = -\frac{8}{27}.\] Keep the negative sign since the two numbers being multiplied have different signs.

Now we have to multiply our result from step 1, \(-\frac{8}{27}\), by \(\frac{1}{2}\):\[-\frac{8}{27} \times \frac{1}{2} = -\frac{8\times1}{27\times2}.\]Calculate the multiplication:- Multiply the numerators: \(8 \times 1 = 8\).- Multiply the denominators: \(27 \times 2 = 54\).So, the fraction becomes:\[-\frac{8}{54}.\]Now, simplify the fraction \(-\frac{8}{54}\):- Find the greatest common divisor (GCD) of 8 and 54, which is 2.- Divide both numerator and denominator by 2: \[-\frac{8\div2}{54\div2} = -\frac{4}{27}.\]

The expression \(\left(-\frac{2}{3}\right)^3 \cdot \frac{1}{2}\) evaluates to \(-\frac{4}{27}\) after simplifying the fractional multiplication.