Remember to follow BODMAS, where exponents and roots follow immediately after Brackets, Example 3: calculate \( \begin{array}{ll}\text { a) } \frac{1}{4}+\frac{1}{3} \times\left(\frac{4}{5}+\frac{1}{10}\right) & \text { b) }-\left(-\frac{1}{2}\right)^{3}-\left(-2 \frac{1}{2}\right)^{2}\end{array} \)

Certainly! Let's solve both parts of Example 3 step by step using the BODMAS/BIDMAS rule, which stands for: - **B**rackets - **O**rders (i.e., powers and roots, etc.) - **D**ivision and **M**ultiplication - **A**ddition and **S**ubtraction ### Example 3: #### a) \( \frac{1}{4} + \frac{1}{3} \times \left( \frac{4}{5} + \frac{1}{10} \right) \) **Step 1: Solve Inside the Brackets** First, calculate the expression inside the brackets: \[ \frac{4}{5} + \frac{1}{10} \] To add these fractions, find a common denominator. The least common denominator (LCD) for 5 and 10 is 10. \[ \frac{4}{5} = \frac{8}{10} \quad \text{(since } 4 \times 2 = 8 \text{ and } 5 \times 2 = 10\text{)} \] Now, add the fractions: \[ \frac{8}{10} + \frac{1}{10} = \frac{9}{10} \] So, the expression becomes: \[ \frac{1}{4} + \frac{1}{3} \times \frac{9}{10} \] **Step 2: Perform the Multiplication** Next, multiply \( \frac{1}{3} \) by \( \frac{9}{10} \): \[ \frac{1}{3} \times \frac{9}{10} = \frac{9}{30} = \frac{3}{10} \quad \text{(simplified by dividing numerator and denominator by 3)} \] Now, the expression is: \[ \frac{1}{4} + \frac{3}{10} \] **Step 3: Perform the Addition** Add \( \frac{1}{4} \) and \( \frac{3}{10} \): To add these fractions, find a common denominator. The LCD for 4 and 10 is 20. \[ \frac{1}{4} = \frac{5}{20} \quad \text{and} \quad \frac{3}{10} = \frac{6}{20} \] Now, add them: \[ \frac{5}{20} + \frac{6}{20} = \frac{11}{20} \] **Answer for part a:** \( \frac{11}{20} \) --- #### b) \( -\left(-\frac{1}{2}\right)^{3} - \left( -2 \frac{1}{2} \right)^{2} \) First, let's interpret the mixed number \( -2 \frac{1}{2} \): \[ -2 \frac{1}{2} = -\frac{5}{2} \] So the expression becomes: \[ -\left(-\frac{1}{2}\right)^{3} - \left(-\frac{5}{2}\right)^{2} \] **Step 1: Calculate Each Term Separately** 1. **First Term:** \( -\left(-\frac{1}{2}\right)^{3} \) - Calculate the exponent: \[ \left(-\frac{1}{2}\right)^{3} = -\frac{1}{8} \] - Apply the negative sign outside: \[ -\left(-\frac{1}{8}\right) = \frac{1}{8} \] 2. **Second Term:** \( -\left(-\frac{5}{2}\right)^{2} \) - Calculate the exponent: \[ \left(-\frac{5}{2}\right)^{2} = \frac{25}{4} \] - Apply the negative sign outside: \[ -\frac{25}{4} \] **Step 2: Combine the Results** Now, combine both terms: \[ \frac{1}{8} - \frac{25}{4} \] To subtract these fractions, find a common denominator. The LCD for 8 and 4 is 8. \[ \frac{25}{4} = \frac{50}{8} \] Now, subtract: \[ \frac{1}{8} - \frac{50}{8} = -\frac{49}{8} \] This can also be expressed as a mixed number: \[ -\frac{49}{8} = -6 \frac{1}{8} \] **Answer for part b:** \( -\frac{49}{8} \) or \( -6 \frac{1}{8} \)