Use the logarithm property ((A)/(B)) = A - B.

Here are the solutions to the problems. 1. (a) Given that m = x and n = 2x - 1, find an expression for ((mn^2)/(100)). Step 1: Use the logarithm property ((A)/(B)) = A - B. ((mn^2)/(100)) = (mn^2) - (100) Step 2: Use the logarithm properties (AB) = A + B and (A^k) = k A. (mn^2) - (100) = m + (n^2) - (100) = m + 2 n - (100) Step 3: Substitute the given values for m and n, and evaluate (100) (assuming base 10 logarithm). x + 2(2x - 1) - 2 Step 4: Simplify the expression. x + 4x - 2 - 2 = 5x - 4 The expression is 5x - 4. 1. (b) Simplify: (1 - sqrt(5))((1)/(5) + sqrt(5)) leaving your answer in the form m + nsqrt(5), where m and n are rational numbers. Step 1: Expand the product using the distributive property (FOIL). (1 - sqrt(5))((1)/(5) + sqrt(5)) = 1 · (1)/(5) + 1 · sqrt(5) - sqrt(5) · (1)/(5) - sqrt(5) · sqrt(5) Step 2: Simplify each term. = (1)/(5) + sqrt(5) - sqrt(5)5 - 5 Step 3: Group the rational terms and the terms containing sqrt(5). = ((1)/(5) - 5) + (sqrt(5) - sqrt(5)5) Step 4: Perform the arithmetic for each group. = ((1)/(5) - (25)/(5)) + (5sqrt(5)5 - sqrt(5)5) = -(24)/(5) + 4sqrt(5)5 Step 5: Express the answer in the form m + nsqrt(5). = -(24)/(5) + (4)/(5)sqrt(5) The simplified expression is -(24)/(5) + (4)/(5)sqrt(5). 2. (a) Construct a multiplication table for modulo 9 on the set \2, 4, 6, 8\. Step 1: Create a table with the given elements as headers for rows and columns. Step 2: Calculate each product a × b 9 for a, b \2, 4, 6, 8\ and fill the table. For example, 2 2 = (2 × 2) 9 = 4 9 = 4. 2 6 = (2 × 6) 9 = 12 9 = 3. The multiplication table is: |c|c|c|c|c| & 2 & 4 & 6 & 8 \\ 2 & 4 & 8 & 3 & 7 \\ 4 & 8 & 7 & 6 & 5 \\ 6 & 3 & 6 & 0 & 3 \\ 8 & 7 & 5 &