Question 2: a) Express 216 as a product of prime factors. Hence evaluate (216)2/3. b) Solve the inequality 1/2(x+1) >= 1/3(2x+2) and illustrate the answer on a number line.

Here is the solution to question 2: a) i) To express 216 as a product of prime factors, we divide it by the smallest prime numbers until we are left with prime factors. Step 1: Find the prime factors of 216. 216 = 2 × 108 108 = 2 × 54 54 = 2 × 27 27 = 3 × 9 9 = 3 × 3 So, 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2^3 × 3^3. The product of prime factors is 2^3 × 3^3. ii) To evaluate (216)^(2)/(3), we use the prime factorization from part (i). Step 2: Substitute the prime factorization into the expression. (216)^(2)/(3) = (2^3 × 3^3)^(2)/(3) Step 3: Apply the exponent rule (ab)^m = a^m b^m and (a^m)^n = a^mn. (2^3 × 3^3)^(2)/(3) = (2^3)^(2)/(3) × (3^3)^(2)/(3) = 2^3 × (2)/(3) × 3^3 × (2)/(3) = 2^2 × 3^2 = 4 × 9 = 36 The value is 36. b) To solve the inequality (1)/(2)(x+1) (1)/(3)(2x+2) and illustrate the answer on a number line. Step 1: Clear the denominators by multiplying both sides by the least common multiple (LCM) of 2 and 3, which is 6. 6 × (1)/(2)(x+1) 6 × (1)/(3)(2x+2) 3(x+1) 2(2x+2) Step 2: Distribute the numbers on both sides. 3x + 3 4x + 4 Step 3: Gather x terms on one side and constant terms on the other. Subtract 3x from both sides: 3 4x - 3x + 4 3 x + 4 Subtract 4 from both sides: 3 - 4 x -1 x This can be rewritten as x -1. Step 4: Illustrate the solution on a number line. The solution x -1 means all real numbers less than or equal to -1. This is represented by a closed circle at -1 and an arrow extending to the left. Number Line: -------------------- \\ -1 0 1 The solution is x -1. 3 done, 2 left today. You're making progress.