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.
ScanSolve AI Answer
Here are the solutions to your math problems: 2. Simplify i) 125 Step 1: Express 125 as a power of its prime factor. 125 = 5^3 Step 2: Apply the logarithm property a^b = b a. 125 = 5^3 = 3 5 The simplified expression is 3 5. ii) 81a^3b^5c^2 ÷ 27b^2c Step 1: Rewrite the division as a fraction. (81a^3b^5c^2)/(27b^2c) Step 2: Simplify the numerical coefficients and apply the exponent rule (x^m)/(x^n) = x^m-n for the variables. (81)/(27) · a^3 · b^5-2 · c^2-1 3a^3b^3c^1 The simplified expression is 3a^3b^3c. 3. Choose the correct in i) 2 5 + 4 - 2 10 Step 1: Apply the power rule for logarithms, b a = a^b. 5^2 + 4 - 10^2 25 + 4 - 100 Step 2: Apply the product rule, a + b = (ab), and the quotient rule, a - b = ((a)/(b)). (25 × 4) - 100 100 - 100 ((100)/(100)) 1 The correct option is A. 1. ii) 125^-(1)/(3) × 10^-1 × 10^2 Step 1: Simplify each term using exponent rules. a^-n = (1)/(a^n) and a^(1)/(n) = [n]a. 125^-(1)/(3) = (1)/(125^1)3 = (1)/([3]125) = (1)/(5) 10^-1 = (1)/(10) 10^2 = 100 Step 2: Multiply the simplified terms. (1)/(5) × (1)/(10) × 100 (100)/(50) 2 It seems there might be a typo in the options provided in the image, as none of them match the calculated answer of 2. Assuming there was a typo and the options were meant to include 2 or if the question intended a different calculation, based on the provided options (A. 1/5, B. 350, C. 1/50, D. 1), none are correct. If the question was 125^-(1)/(3) × 10^-1 × 10^1, the answer would be 1/5. Given the options, there might be an error in the question or the options. However, if we strictly follow the calculation, the result is 2. Let's re-evaluate the options provided in the image. The options are A. 1/5, B. 350, C. 1/50, D. 1. My calculation gives 2. Since 2 is not among the options, I will state the calculated answer. The simplified expression is 2. 4. Evaluate (1)/(2) 9 Step 1: Apply the power rule for logarithms, b a = a^b. (1)/(2) 9 = 9^(1)/(2) Step 2: Simplify the exponent. sqrt(9) 3 The evaluated expression is 3. iii) Write the log form 5^x = 125 Step 1: Convert the exponential equation to its logarithmic form. If b^y = x, then _b x = y. 5^x = 125 _5 125 = x The logarithmic form is _5 125 = x. 5. Simplify indice ((2)/(3))^x-1 = (27)/(8) Step 1: Express the right side of the equation with a base related to (2)/(3). (27)/(8) = ((3)/(2))^3 Step 2: Rewrite ((3)/(2))^3 as a power of (2)/(3) using the rule ((a)/(b))^-n = ((b)/(a))^n. ((3)/(2))^3 = ((2)/(3))^-3 Step 3: Equate the bases and then the exponents. ((2)/(3))^x-1 = ((2)/(3))^-3 x-1 = -3 Step 4: Solve for x. x = -3 + 1 x = -2 The value of x is -2. What's next? 📸