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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Let's go through these problems step by step. Multiplication of Algebraic Expressions a) (5a^2)(4a^2) Step 1: Multiply the numerical coefficients. 5 × 4 = 20 Step 2: Multiply the variable parts. When multiplying variables with the same base, add their exponents (e.g., a^m · a^n = a^m+n). a^2 · a^2 = a^2+2 = a^4 Step 3: Combine the results from Step 1 and Step 2. 20a^4 The final answer is 20a^4. b) (-5xy^2)(4x^2y^3) Step 1: Multiply the numerical coefficients, including their signs. -5 × 4 = -20 Step 2: Multiply the x variables. Remember that x is x^1. x^1 · x^2 = x^1+2 = x^3 Step 3: Multiply the y variables. y^2 · y^3 = y^2+3 = y^5 Step 4: Combine the results from Step 1, Step 2, and Step 3. -20x^3y^5 The final answer is -20x^3y^5. c) (3x - 2y^2z - 3z^2) × (2x^2yz^2) Step 1: Distribute the monomial 2x^2yz^2 to each term inside the parenthesis. (3x)(2x^2yz^2) - (2y^2z)(2x^2yz^2) - (3z^2)(2x^2yz^2) Step 2: Multiply each pair of terms. For the first term: (3x)(2x^2yz^2) = (3 × 2)(x^1 · x^2)(y)(z^2) = 6x^1+2yz^2 = 6x^3yz^2 For the second term: -(2y^2z)(2x^2yz^2) = -(2 × 2)(x^2)(y^2 · y^1)(z^1 · z^2) = -4x^2y^2+1z^1+2 = -4x^2y^3z^3 For the third term: -(3z^2)(2x^2yz^2) = -(3 × 2)(x^2)(y)(z^2 · z^2) = -6x^2yz^2+2 = -6x^2yz^4 Step 3: Combine the results. 6x^3yz^2 - 4x^2y^3z^3 - 6x^2yz^4 The final answer is 6x^3yz^2 - 4x^2y^3z^3 - 6x^2yz^4. Question One: Simplify the following 1. 7(5a^2b + 7x - 2a^2b) Step 1: Simplify the terms inside the parenthesis first by combining like terms. The like terms are 5a^2b and -2a^2b. 5a^2b - 2a^2b = (5-2)a^2b = 3a^2b So the expression inside the parenthesis becomes: (3a^2b + 7x) Step 2: Distribute the 7 to each term inside the simplified parenthesis. 7(3a^2b) + 7(7x) Step 3: Perform the multiplication. 21a^2b + 49x The final answer is 21a^2b + 49x. 2. -6y(p^2 - 3y + 7 - 5y) Step 1: Simplify the terms inside the parenthesis first by combining like terms. The like terms are -3y and -5y. -3y - 5y = (-3-5)y = -8y So the expression inside the parenthesis becomes: (p^2 - 8y + 7) Step 2: Distribute the -6y to each term inside the simplified parenthesis. -6y(p^2) - 6y(-8y) - 6y(7) Step 3: Perform the multiplication for each term. -6y(p^2) = -6p^2y -6y(-8y) = (-6 × -8)(y · y) = 48y^2 -6y(7) = -42y Step 4: Combine the results. -6p^2y + 48y^2 - 42y The final answer is -6p^2y + 48y^2 - 42y. That's 2 down. 3 left today — send the next one.