This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Welcome back nikoiwen3@gmail.com688_82 — been a while! Let's pick up where you left off. QUESTION 1. Use the properties of indices to evaluate the following, giving the answers as fractions or whole numbers. a) 6(3^-1) + 3^0 Step 1: Evaluate the terms with exponents. We use the properties a^-n = (1)/(a^n) and a^0 = 1. 3^-1 = (1)/(3) 3^0 = 1 Step 2: Substitute these values into the expression. 6((1)/(3)) + 1 Step 3: Perform the multiplication. 2 + 1 Step 4: Perform the addition. 3 The final answer is 3. b) 3^-1 × 6^2 + 6^0 Step 1: Evaluate the terms with exponents. 3^-1 = (1)/(3) 6^2 = 36 6^0 = 1 Step 2: Substitute these values into the expression. (1)/(3) × 36 + 1 Step 3: Perform the multiplication. 12 + 1 Step 4: Perform the addition. 13 The final answer is 13. QUESTION 2. Calculate the following giving the answers as fractions. a) 3^-22^-2 - 2^-33^-3 Step 1: Evaluate the terms with negative exponents. 3^-2 = (1)/(3^2) = (1)/(9) 2^-2 = (1)/(2^2) = (1)/(4) 2^-3 = (1)/(2^3) = (1)/(8) 3^-3 = (1)/(3^3) = (1)/(27) Step 2: Substitute these values into the expression. (1)/(9)(1)/(4) - (1)/(8)(1)/(27) Step 3: Simplify the complex fractions. (1)/(9) × (4)/(1) - (1)/(8) × (27)/(1) = (4)/(9) - (27)/(8) Step 4: Find a common denominator and subtract the fractions. The least common multiple of 9 and 8 is 72. (4 × 8)/(9 × 8) - (27 × 9)/(8 × 9) = (32)/(72) - (243)/(72) Step 5: Perform the subtraction. (32 - 243)/(72) = (-211)/(72) The final answer is -(211)/(72). b) 5(2^3)^-1 + 4^-2 Step 1: Evaluate the terms with exponents. 2^3 = 8 (2^3)^-1 = 8^-1 = (1)/(8) 4^-2 = (1)/(4^2) = (1)/(16) Step 2: Substitute these values into the expression. 5((1)/(8)) + (1)/(16) Step 3: Perform the multiplication. (5)/(8) + (1)/(16) Step 4: Find a common denominator and add the fractions. The least common multiple of 8 and 16 is 16. (5 × 2)/(8 × 2) + (1)/(16) = (10)/(16) + (1)/(16) Step 5: Perform the addition. (10 + 1)/(16) = (11)/(16) The final answer is (11)/(16). Send me the next one 📸