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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3: 1. Write down the number of terms in each expression. a) 3 + 2ab - 4ac + 6bc Step 1: Identify terms separated by addition or subtraction. The terms are 3, 2ab, -4ac, and 6bc. There are 4 terms. b) 6df - 7 + 9b × 2c Step 1: Simplify any multiplication within a term. 9b × 2c = 18bc. The expression becomes 6df - 7 + 18bc. Step 2: Identify terms separated by addition or subtraction. The terms are 6df, -7, and 18bc. There are 3 terms. c) (6a - 4b) + (10c + 12d) Step 1: Remove the parentheses. The expression becomes 6a - 4b + 10c + 12d. Step 2: Identify terms separated by addition or subtraction. The terms are 6a, -4b, 10c, and 12d. There are 4 terms. d) (a^2 - d) × (c^3 - 2d) Step 1: Recognize that this is a single product. The entire expression is one multiplication operation. There is 1 term. 2. Are 3x^2y^3 and -5x^3y^2 like terms? a) 3x^2y^3 and -5x^3y^2 Step 1: Compare the variables and their exponents. For 3x^2y^3, the variables are x^2y^3. For -5x^3y^2, the variables are x^3y^2. Since the exponents for x (2 vs 3) and y (3 vs 2) are not the same, they are not like terms. The answer is No. b) 4a^2b^3 and -2b^3a^2 Step 1: Compare the variables and their exponents. For 4a^2b^3, the variables are a^2b^3. For -2b^3a^2, the variables are b^3a^2. Since a^2b^3 is equivalent to b^3a^2 (multiplication is commutative), the variables and their exponents are the same. The answer is Yes. 3. Write down the constant terms in each of these expressions. a) 6x^7 - 10xy + 5 - 7y^3 Step 1: Identify the term that does not contain any variables. The constant term is 5. The answer is 5. b) 12 + 4abc - 2ba + 13cd Step 1: Identify the term that does not contain any variables. The constant term is 12. The answer is 12. c) 24m^3n + 12mn^2 - 33 - 60m^2n^3 Step 1: Identify the term that does not contain any variables. The constant term is -33. The answer is -33. 4. Simplify where possible. a) y^2 + y^2 Step 1: Combine the like terms. y^2 + y^2 = 1y^2 + 1y^2 = (1+1)y^2 = 2y^2 The simplified expression is 2y^2. b) 3y^2 - 10y^2 Step 1: Combine the like terms. 3y^2 - 10y^2 = (3-10)y^2 = -7y^2 The simplified expression is -7y^2. c) 2x^2 + 9x - x^2 + 12x Step 1: Group the like terms. (2x^2 - x^2) + (9x + 12x) Step 2: Combine the like terms. (2-1)x^2 + (9+12)x = 1x^2 + 21x = x^2 + 21x The simplified expression is x^2 + 21x. d) 15ab + ab Step 1: Combine the like terms. 15ab + 1ab = (15+1)ab = 16ab The simplified expression is 16ab. e) 3a + 4b - 6c - 2a - 4b + 5c Step 1: Group the like terms. (3a - 2a) + (4b - 4b) + (-6c + 5c) Step 2: Combine the like terms. (3-2)a + (4-4)b + (-6+5)c = 1a + 0b - 1c = a - c The simplified expression is a - c. f) 10x^2y^3 - 5x^2y^3 Step 1: Combine the like terms. 10x^2y^3 - 5x^2y^3 = (10-5)x^2y^3 = 5x^2y^3 The simplified expression is 5x^2y^3. That's 2 down. 3 left today — send the next one.