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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Here are the solutions to the problems: 1. Simplify: a) 16^(1)/(2) × 27^(1)/(3) Step 1: Evaluate each term. 16^(1)/(2) = sqrt(16) = 4 27^(1)/(3) = [3]27 = 3 Step 2: Multiply the results. 4 × 3 = 12 The simplified expression is 12. b) 8^(5)/(3) × 49^(3)/(2) Step 1: Evaluate each term using the property a^(m)/(n) = ([n]a)^m. 8^(5)/(3) = ([3]8)^5 = (2)^5 = 32 49^(3)/(2) = (sqrt(49))^3 = (7)^3 = 343 Step 2: Multiply the results. 32 × 343 = 10976 The simplified expression is 10976. 2. Use logarithms to work out: a) 3/(8.237 × 0.0045) Let X = 3/(8.237 × 0.0045). Taking the logarithm of both sides: X = (1)/(3) [ (21.8) + (0.1275) - ((8.237) + (0.0045)) ] Step 1: Find the logarithms of the individual numbers. (21.8) = 1.3385 (0.1275) = 1.1055 (8.237) = 0.9158 (0.0045) = 3.6532 Step 2: Calculate the logarithm of the numerator. (numerator) = (21.8) + (0.1275) = 1.3385 + 1.1055 = (1 + 1) + (0.3385 + 0.1055) = 0 + 0.4440 = 0.4440 Step 3: Calculate the logarithm of the denominator. (denominator) = (8.237) + (0.0045) = 0.9158 + 3.6532 = (0 + 3) + (0.9158 + 0.6532) = -3 + 1.5690 = -2 + 0.5690 = 2.5690 Step 4: Calculate the logarithm of the quotient. (numeratordenominator) = (numerator) - (denominator) = 0.4440 - 2.5690 = 0.4440 - (-2 + 0.5690) = 0.4440 + 2 - 0.5690 = 2.4440 - 0.5690 = 1.8750 Step 5: Calculate X. X = (1)/(3) (1.8750) = 0.6250 Step 6: Find the antilogarithm of 0.6250. X = antilog(0.6250) ≈ 4.217 The value is 4.217. b) 5/(2387 × 2.883) Let Y = 5/(2387 × 2.883). Taking the logarithm of both sides: Y = (1)/(5) [ (0.4583) - ((2387) + (2.883)) ] Step 1: Find the logarithms of the individual numbers. (0.4583) = 1.6612 (2387) = 3.3778 (2.883) = 0.4598 Step 2: Calculate the logarithm of the denominator. (denominator) = (2387) + (2.883) = 3.3778 + 0.4598 = 3.8376 Step 3: Calculate the logarithm of the quotient. (numeratordenominator) = (0.4583) - (denominator) = 1.6612 - 3.8376 = (-1 + 0.6612) - 3.8376 = -1 - 3.8376 + 0.6612 = -4.8376 + 0.6612 = -4.1764 Step 4: Prepare the logarithm for division by 5. To make the characteristic divisible by 5, we write -4.1764 as -5 + 0.8236 = 5.8236. Step 5