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 for removing the brackets and collecting like terms. a) Remove the brackets and collect the like terms for 4a^2(2a+b) - 3a^2(5a+2b). Step 1: Distribute 4a^2 and -3a^2 into their respective parentheses. 4a^2(2a+b) - 3a^2(5a+2b) = (4a^2 · 2a) + (4a^2 · b) - (3a^2 · 5a) - (3a^2 · 2b) = 8a^3 + 4a^2b - 15a^3 - 6a^2b Step 2: Collect like terms. = (8a^3 - 15a^3) + (4a^2b - 6a^2b) = -7a^3 - 2a^2b The simplified expression is -7a^3 - 2a^2b. b) Remove the brackets and collect the like terms for 3m(m^2 - 2n^2) + 6m(m^2 - 2n^2). Step 1: Distribute 3m and 6m into their respective parentheses. 3m(m^2 - 2n^2) + 6m(m^2 - 2n^2) = (3m · m^2) - (3m · 2n^2) + (6m · m^2) - (6m · 2n^2) = 3m^3 - 6mn^2 + 6m^3 - 12mn^2 Step 2: Collect like terms. = (3m^3 + 6m^3) + (-6mn^2 - 12mn^2) = 9m^3 - 18mn^2 The simplified expression is 9m^3 - 18mn^2. c) Remove the brackets and collect the like terms for 4y^2(x^2 - 4xy + y^2) - 4y^2(x^2 + 4xy - y^2). Step 1: Distribute 4y^2 and -4y^2 into their respective parentheses. 4y^2(x^2 - 4xy + y^2) - 4y^2(x^2 + 4xy - y^2) = (4y^2 · x^2) - (4y^2 · 4xy) + (4y^2 · y^2) - (4y^2 · x^2) - (4y^2 · 4xy) + (4y^2 · y^2) = 4x^2y^2 - 16xy^3 + 4y^4 - 4x^2y^2 - 16xy^3 + 4y^4 Step 2: Collect like terms. = (4x^2y^2 - 4x^2y^2) + (-16xy^3 - 16xy^3) + (4y^4 + 4y^4) = 0 - 32xy^3 + 8y^4 = -32xy^3 + 8y^4 The simplified expression is -32xy^3 + 8y^4. d) Remove the brackets and collect the like terms for 4x(x^2 - 4x + 6) + 2x(5x^2 + 3x - 1). Step 1: Distribute 4x and 2x into their respective parentheses. 4x(x^2 - 4x + 6) + 2x(5x^2 + 3x - 1) = (4x · x^2) - (4x · 4x) + (4x · 6) + (2x · 5x^2) + (2x · 3x) - (2x · 1) = 4x^3 - 16x^2 + 24x + 10x^3 + 6x^2 - 2x Step 2: Collect like terms. = (4x^3 + 10x^3) + (-16x^2 + 6x^2) + (24x - 2x) = 14x^3 - 10x^2 + 22x The simplified expression is 14x^3 - 10x^2 + 22x. e) Remove the brackets and collect the like terms for p^2(2p^2 - 5b^2) - 2(p^4 + 3b^4) + 3b^2(5p^2 - 2b^2). Step 1: Distribute p^2, -2, and 3b^2 into their respective parentheses. p^2(2p^2 - 5b^2) - 2(p^4 + 3b^4) + 3b^2(5p^2 - 2b^2) = (p^2 · 2p^2) - (p^2 · 5b^2) - (2 · p^4) - (2 · 3b^4) + (3b^2 · 5p^2) - (3b^2 · 2b^2) = 2p^4 - 5p^2b^2 - 2p^4 - 6b^4 + 15p^2b^2 - 6b^4 Step 2: Collect like terms. = (2p^4 - 2p^4) + (-5p^2b^2 + 15p^2b^2) + (-6b^4 - 6b^4) = 0 + 10p^2b^2 - 12b^4 = 10p^2b^2 - 12b^4 The simplified expression is 10p^2b^2 - 12b^4. f) Remove the brackets and collect the like terms for 10(k - (1)/(5)) + 8(k + (1)/(2)) - 9(k - (1)/(3)). Step 1: Distribute 10, 8, and -9 into their respective parentheses. 10(k - (1)/(5)) + 8(k + (1)/(2)) - 9(k - (1)/(3)) = (10 · k) - (10 · (1)/(5)) + (8 · k) + (8 · (1)/(2)) - (9 · k) - (9 · -(1)/(3)) = 10k - (10)/(5) + 8k + (8)/(2) - 9k + (9)/(3) = 10k - 2 + 8k + 4 - 9k + 3 Step 2: Collect like terms. = (10k + 8k - 9k) + (-2 + 4 + 3) = (18k - 9k) + (2 + 3) = 9k + 5 The simplified expression is 9k + 5. That's 2 down. 3 left today — send the next one.