ay not use a calculator for this ex (2) \( \left(\frac{1}{2}\right)^{3} \) (5) \( \left(-\frac{2}{3}\right)^{2} \) (8) \( \left(1 \frac{1}{3}\right)^{3} \)

Below is the step-by-step solution for each part. --- ### Problem (2) Evaluate \( \left(\frac{1}{2}\right)^{3} \). 1. Recall that raising a fraction to the 3rd power means multiplying the fraction by itself three times: \[ \left(\frac{1}{2}\right)^3 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}. \] 2. Multiply the numerators together and the denominators together: \[ \frac{1 \times 1 \times 1}{2 \times 2 \times 2} = \frac{1}{8}. \] **Answer for (2):** \(\frac{1}{8}\) --- ### Problem (5) Evaluate \( \left(-\frac{2}{3}\right)^{2} \). 1. Note that squaring a number means multiplying it by itself: \[ \left(-\frac{2}{3}\right)^2 = \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right). \] 2. The product of two negative numbers is positive. Multiply the numerators and the denominators: \[ \frac{(-2) \times (-2)}{3 \times 3} = \frac{4}{9}. \] **Answer for (5):** \(\frac{4}{9}\) --- ### Problem (8) Evaluate \( \left(1 \frac{1}{3}\right)^{3} \). 1. First, convert the mixed number \(1 \frac{1}{3}\) to an improper fraction: \[ 1 \frac{1}{3} = \frac{3 \times 1 + 1}{3} = \frac{4}{3}. \] 2. Now, cube the fraction: \[ \left(\frac{4}{3}\right)^3 = \frac{4^3}{3^3}. \] 3. Calculate the powers: \[ 4^3 = 64 \quad \text{and} \quad 3^3 = 27. \] 4. Thus: \[ \left(\frac{4}{3}\right)^3 = \frac{64}{27}. \] **Answer for (8):** \(\frac{64}{27}\) --- These are the step-by-step solutions without using a calculator.