Algebra Products

1. Problem: Find the product $2a^3(3a + 5b)$. Formula: Use distributive property $x(y+z) = xy + xz$. Step 1: Multiply $2a^3$ by each term inside the parentheses: $$2a^3 \times 3a = 6a^4$$ $$2a^3 \times 5b = 10a^3b$$ Step 2: Combine results: $$6a^4 + 10a^3b$$ Answer: $6a^4 + 10a^3b$ --- 2. Problem: Find the product $-11a(3a + 2b)$. Step 1: Multiply $-11a$ by each term: $$-11a \times 3a = -33a^2$$ $$-11a \times 2b = -22ab$$ Step 2: Combine: $$-33a^2 - 22ab$$ Answer: $-33a^2 - 22ab$ --- 3. Problem: Find the product $-5a(7a - 2b)$. Step 1: Multiply $-5a$ by each term: $$-5a \times 7a = -35a^2$$ $$-5a \times (-2b) = 10ab$$ Step 2: Combine: $$-35a^2 + 10ab$$ Answer: $-35a^2 + 10ab$ --- 4. Problem: Find the product $-11y^2(3y + 7)$. Step 1: Multiply $-11y^2$ by each term: $$-11y^2 \times 3y = -33y^3$$ $$-11y^2 \times 7 = -77y^2$$ Step 2: Combine: $$-33y^3 - 77y^2$$ Answer: $-33y^3 - 77y^2$ --- 5. Problem: Find the product $\frac{6x}{5}(x^3 + y^3)$. Step 1: Multiply $\frac{6x}{5}$ by each term: $$\frac{6x}{5} \times x^3 = \frac{6x^4}{5}$$ $$\frac{6x}{5} \times y^3 = \frac{6xy^3}{5}$$ Step 2: Combine: $$\frac{6x^4}{5} + \frac{6xy^3}{5}$$ Answer: $\frac{6x^4}{5} + \frac{6xy^3}{5}$ --- 6. Problem: Find the product $xy(x^3 - y^3)$. Step 1: Multiply $xy$ by each term: $$xy \times x^3 = x^4y$$ $$xy \times (-y^3) = -xy^4$$ Step 2: Combine: $$x^4y - xy^4$$ Answer: $x^4y - xy^4$ --- 7. Problem: Find the product $0.1y(0.1x^5 + 0.1y)$. Step 1: Multiply $0.1y$ by each term: $$0.1y \times 0.1x^5 = 0.01x^5y$$ $$0.1y \times 0.1y = 0.01y^2$$ Step 2: Combine: $$0.01x^5y + 0.01y^2$$ Answer: $0.01x^5y + 0.01y^2$ --- 8. Problem: Simplify $\frac{7}{4} ab^2c - \frac{6}{25} a^2 c^2 (-50a^2 b^2 c^2)$. Step 1: Multiply inside the second term: $$- \frac{6}{25} a^2 c^2 \times (-50a^2 b^2 c^2) = \frac{6}{25} \times 50 a^{2+2} b^2 c^{2+2} = \frac{6 \times 50}{25} a^4 b^2 c^4 = 12 a^4 b^2 c^4$$ Step 2: Combine terms: $$\frac{7}{4} ab^2 c + 12 a^4 b^2 c^4$$ Answer: $\frac{7}{4} ab^2 c + 12 a^4 b^2 c^4$ --- 9. Problem: Simplify $- \frac{8}{27} xyz \left( \frac{3}{2} xy^2 - \frac{9}{4} xy^2 z^3 \right)$. Step 1: Multiply $- \frac{8}{27} xyz$ by each term: $$- \frac{8}{27} xyz \times \frac{3}{2} xy^2 = - \frac{8}{27} \times \frac{3}{2} x^{1+1} y^{1+2} z^{1} = - \frac{24}{54} x^2 y^3 z = - \frac{12}{27} x^2 y^3 z = - \frac{4}{9} x^2 y^3 z$$ $$- \frac{8}{27} xyz \times \left(- \frac{9}{4} xy^2 z^3 \right) = + \frac{8}{27} \times \frac{9}{4} x^{1+1} y^{1+2} z^{1+3} = \frac{72}{108} x^2 y^3 z^4 = \frac{2}{3} x^2 y^3 z^4$$ Step 2: Combine: $$- \frac{4}{9} x^2 y^3 z + \frac{2}{3} x^2 y^3 z^4$$ Answer: $- \frac{4}{9} x^2 y^3 z + \frac{2}{3} x^2 y^3 z^4$ --- 10. Problem: Simplify $\frac{7}{27} xy \left( \frac{3}{2} y^2 z - \frac{9}{4} xy^2 \right)$. Step 1: Multiply $\frac{7}{27} xy$ by each term: $$\frac{7}{27} xy \times \frac{3}{2} y^2 z = \frac{7}{27} \times \frac{3}{2} x y^{1+2} z = \frac{21}{54} x y^3 z = \frac{7}{18} x y^3 z$$ $$\frac{7}{27} xy \times \left(- \frac{9}{4} xy^2 \right) = - \frac{7}{27} \times \frac{9}{4} x^{1+1} y^{1+2} = - \frac{63}{108} x^2 y^3 = - \frac{21}{36} x^2 y^3 = - \frac{7}{12} x^2 y^3$$ Step 2: Combine: $$\frac{7}{18} x y^3 z - \frac{7}{12} x^2 y^3$$ Answer: $\frac{7}{18} x y^3 z - \frac{7}{12} x^2 y^3$ --- 11. Problem: Find the product $1.5x(10x^2 y - 100xy^2)$. Step 1: Multiply $1.5x$ by each term: $$1.5x \times 10x^2 y = 15 x^3 y$$ $$1.5x \times (-100xy^2) = -150 x^2 y^2$$ Step 2: Combine: $$15 x^3 y - 150 x^2 y^2$$ Answer: $15 x^3 y - 150 x^2 y^2$ --- 12. Problem: Find the product $4.1xy(1.1x - y)$. Step 1: Multiply $4.1xy$ by each term: $$4.1xy \times 1.1x = 4.51 x^2 y$$ $$4.1xy \times (-y) = -4.1 x y^2$$ Step 2: Combine: $$4.51 x^2 y - 4.1 x y^2$$ Answer: $4.51 x^2 y - 4.1 x y^2$ --- 13. Problem: Find the product $250.5xy (x^2 + \frac{y}{10})$. Step 1: Multiply $250.5xy$ by each term: $$250.5xy \times x^2 = 250.5 x^3 y$$ $$250.5xy \times \frac{y}{10} = 25.05 x y^2$$ Step 2: Combine: $$250.5 x^3 y + 25.05 x y^2$$ Answer: $250.5 x^3 y + 25.05 x y^2$ --- 14. Problem: Find the product $\frac{7}{5} x^2 y \left( \frac{3}{5} x^2 + \frac{2}{5} x \right)$. Step 1: Multiply $\frac{7}{5} x^2 y$ by each term: $$\frac{7}{5} x^2 y \times \frac{3}{5} x^2 = \frac{21}{25} x^{4} y$$ $$\frac{7}{5} x^2 y \times \frac{2}{5} x = \frac{14}{25} x^{3} y$$ Step 2: Combine: $$\frac{21}{25} x^{4} y + \frac{14}{25} x^{3} y$$ Answer: $\frac{21}{25} x^{4} y + \frac{14}{25} x^{3} y$ --- 15. Problem: Simplify $\frac{4}{3} a^2 + b^2 - 3c^2$. This is already simplified as a sum of terms. Answer: $\frac{4}{3} a^2 + b^2 - 3 c^2$ --- 16. Problem: Find the product $24x^2 (1 - 2x)$ and evaluate for $x=3$. Step 1: Multiply: $$24x^2 \times 1 = 24x^2$$ $$24x^2 \times (-2x) = -48 x^3$$ Step 2: Combine: $$24x^2 - 48 x^3$$ Step 3: Evaluate at $x=3$: $$24(3)^2 - 48(3)^3 = 24 \times 9 - 48 \times 27 = 216 - 1296 = -1080$$ Answer: $24x^2 - 48 x^3$, value at $x=3$ is $-1080$ --- 17. Problem: Find the product $-3y(xy + y^2)$ and evaluate for $x=4$, $y=5$. Step 1: Multiply: $$-3y \times xy = -3 x y^2$$ $$-3y \times y^2 = -3 y^3$$ Step 2: Combine: $$-3 x y^2 - 3 y^3$$ Step 3: Evaluate: $$-3 \times 4 \times 5^2 - 3 \times 5^3 = -3 \times 4 \times 25 - 3 \times 125 = -300 - 375 = -675$$ Answer: $-3 x y^2 - 3 y^3$, value is $-675$ --- 18. Problem: Multiply $\frac{3}{2} x^2 y^3$ by $(2x - y)$ and verify for $x=1$, $y=2$. Step 1: Multiply: $$\frac{3}{2} x^2 y^3 \times 2x = 3 x^3 y^3$$ $$\frac{3}{2} x^2 y^3 \times (-y) = - \frac{3}{2} x^2 y^4$$ Step 2: Combine: $$3 x^3 y^3 - \frac{3}{2} x^2 y^4$$ Step 3: Evaluate: $$3 \times 1^3 \times 2^3 - \frac{3}{2} \times 1^2 \times 2^4 = 3 \times 8 - \frac{3}{2} \times 16 = 24 - 24 = 0$$ Answer: $3 x^3 y^3 - \frac{3}{2} x^2 y^4$, value is $0$ --- 19. Problem: Multiply and evaluate for $x=-1$, $y=0.25$, $z=0.05$: (i) $15 y^2 (2 - 3x)$ Step 1: Multiply inside: $$15 y^2 \times 2 = 30 y^2$$ $$15 y^2 \times (-3x) = -45 x y^2$$ Step 2: Combine: $$30 y^2 - 45 x y^2$$ Step 3: Evaluate: $$30 (0.25)^2 - 45 (-1)(0.25)^2 = 30 \times 0.0625 + 45 \times 0.0625 = 1.875 + 2.8125 = 4.6875$$ (ii) $-3x (y^2 + z^2)$ Step 1: Inside parentheses: $$y^2 + z^2 = (0.25)^2 + (0.05)^2 = 0.0625 + 0.0025 = 0.065$$ Step 2: Multiply: $$-3 \times (-1) \times 0.065 = 0.195$$ (iii) $z^2 (x - y)$ Step 1: Calculate: $$z^2 = (0.05)^2 = 0.0025$$ $$x - y = -1 - 0.25 = -1.25$$ Step 2: Multiply: $$0.0025 \times (-1.25) = -0.003125$$ (iv) $xz (x^2 + y^2)$ Step 1: Calculate: $$x z = -1 \times 0.05 = -0.05$$ $$x^2 + y^2 = (-1)^2 + (0.25)^2 = 1 + 0.0625 = 1.0625$$ Step 2: Multiply: $$-0.05 \times 1.0625 = -0.053125$$ Answer: (i) $30 y^2 - 45 x y^2 = 4.6875$ (ii) $-3x (y^2 + z^2) = 0.195$ (iii) $z^2 (x - y) = -0.003125$ (iv) $xz (x^2 + y^2) = -0.053125$ --- 20. Problem: Simplify each expression: (i) $2x^2 (x^3 - x) - 3x (x^4 + 2x) - 2 (x^4 - 3x^2)$ Step 1: Expand: $$2x^2 \times x^3 = 2x^5$$ $$2x^2 \times (-x) = -2x^3$$ $$-3x \times x^4 = -3x^5$$ $$-3x \times 2x = -6x^2$$ $$-2 \times x^4 = -2x^4$$ $$-2 \times (-3x^2) = +6x^2$$ Step 2: Combine all: $$2x^5 - 2x^3 - 3x^5 - 6x^2 - 2x^4 + 6x^2$$ Step 3: Simplify like terms: $$2x^5 - 3x^5 = -x^5$$ $$-6x^2 + 6x^2 = 0$$ Final: $$-x^5 - 2x^4 - 2x^3$$ (ii) $x^3 y (x^2 - 2x) + 2xy (x^3 - x^4)$ Step 1: Expand: $$x^3 y \times x^2 = x^5 y$$ $$x^3 y \times (-2x) = -2 x^4 y$$ $$2 x y \times x^3 = 2 x^4 y$$ $$2 x y \times (-x^4) = -2 x^5 y$$ Step 2: Combine: $$x^5 y - 2 x^4 y + 2 x^4 y - 2 x^5 y = (x^5 y - 2 x^5 y) + (-2 x^4 y + 2 x^4 y) = - x^5 y + 0 = - x^5 y$$ (iii) $3a^2 + 2 (a + 2) - 3a (2a + 1)$ Step 1: Expand: $$2 (a + 2) = 2a + 4$$ $$-3a (2a + 1) = -6a^2 - 3a$$ Step 2: Combine all: $$3a^2 + 2a + 4 - 6a^2 - 3a = (3a^2 - 6a^2) + (2a - 3a) + 4 = -3a^2 - a + 4$$ (iv) $x (x + 4) + 3x (2x^2 - 1) + 4x^2 + 4$ Step 1: Expand: $$x (x + 4) = x^2 + 4x$$ $$3x (2x^2 - 1) = 6x^3 - 3x$$ Step 2: Combine all: $$x^2 + 4x + 6x^3 - 3x + 4x^2 + 4 = 6x^3 + (x^2 + 4x^2) + (4x - 3x) + 4 = 6x^3 + 5x^2 + x + 4$$ (v) $a (b - c) - b (c - a) - c (a - b)$ Step 1: Expand: $$a b - a c - b c + a b - c a + c b$$ Step 2: Combine like terms: $$a b + a b = 2 a b$$ $$- a c - c a = - 2 a c$$ $$- b c + c b = 0$$ Answer: $$2 a b - 2 a c$$ (vi) $a (b - c) + b (c - a) + c (a - b)$ Step 1: Expand: $$a b - a c + b c - b a + c a - c b$$ Step 2: Combine like terms: $$a b - b a = 0$$ $$- a c + c a = 0$$ $$b c - c b = 0$$ Answer: $0$ (vii) $4ab (a - b) - 6a^2 (b - b^2) - 3b^2 (2a^2 - a) + 2ab (b - a)$ Step 1: Expand each: $$4ab (a - b) = 4a^2 b - 4ab^2$$ $$-6a^2 (b - b^2) = -6a^2 b + 6a^2 b^2$$ $$-3b^2 (2a^2 - a) = -6a^2 b^2 + 3a b^2$$ $$2ab (b - a) = 2ab^2 - 2a^2 b$$ Step 2: Combine all: $$(4a^2 b - 4ab^2) + (-6a^2 b + 6a^2 b^2) + (-6a^2 b^2 + 3a b^2) + (2ab^2 - 2a^2 b)$$ Step 3: Group like terms: $$a^2 b: 4a^2 b - 6a^2 b - 2a^2 b = (4 - 6 - 2) a^2 b = -4 a^2 b$$ $$ab^2: -4ab^2 + 6a^2 b^2 - 6a^2 b^2 + 3a b^2 + 2ab^2 = (-4 + 3 + 2) ab^2 = 1 ab^2$$ Answer: $-4 a^2 b + ab^2$ (viii) $x^2(x^2 + 1) - x^3(x + 1) - x(x^3 - x)$ Step 1: Expand: $$x^2 \times x^2 = x^4$$ $$x^2 \times 1 = x^2$$ $$- x^3 \times x = - x^4$$ $$- x^3 \times 1 = - x^3$$ $$- x \times x^3 = - x^4$$ $$- x \times (- x) = + x^2$$ Step 2: Combine all: $$x^4 + x^2 - x^4 - x^3 - x^4 + x^2 = (x^4 - x^4 - x^4) + (x^2 + x^2) - x^3 = - x^4 + 2 x^2 - x^3$$ Answer: $- x^4 - x^3 + 2 x^2$ (ix) $2a^2 + 3a(1 - 2a^3) + a(a + 1)$ Step 1: Expand: $$3a \times 1 = 3a$$ $$3a \times (-2a^3) = -6 a^4$$ $$a \times a = a^2$$ $$a \times 1 = a$$ Step 2: Combine all: $$2a^2 + 3a - 6 a^4 + a^2 + a = -6 a^4 + (2a^2 + a^2) + (3a + a) = -6 a^4 + 3 a^2 + 4 a$$ Answer: $-6 a^4 + 3 a^2 + 4 a$ (x) $a^2(2a - 1) + 3a + a^3 - 8$ Step 1: Expand: $$a^2 \times 2a = 2 a^3$$ $$a^2 \times (-1) = - a^2$$ Step 2: Combine all: $$2 a^3 - a^2 + 3 a + a^3 - 8 = (2 a^3 + a^3) - a^2 + 3 a - 8 = 3 a^3 - a^2 + 3 a - 8$$ Answer: $3 a^3 - a^2 + 3 a - 8$ (xi) $\frac{3}{2} x^2 (x^2 - 1) + \frac{1}{4} x^2 (x^2 + x) - \frac{3}{4} x (x^3 - 1)$ Step 1: Expand: $$\frac{3}{2} x^2 \times x^2 = \frac{3}{2} x^4$$ $$\frac{3}{2} x^2 \times (-1) = - \frac{3}{2} x^2$$ $$\frac{1}{4} x^2 \times x^2 = \frac{1}{4} x^4$$ $$\frac{1}{4} x^2 \times x = \frac{1}{4} x^3$$ $$- \frac{3}{4} x \times x^3 = - \frac{3}{4} x^4$$ $$- \frac{3}{4} x \times (-1) = + \frac{3}{4} x$$ Step 2: Combine all: $$\left( \frac{3}{2} x^4 + \frac{1}{4} x^4 - \frac{3}{4} x^4 \right) + \frac{1}{4} x^3 + \left( - \frac{3}{2} x^2 \right) + \frac{3}{4} x$$ Step 3: Simplify coefficients: $$\frac{3}{2} + \frac{1}{4} - \frac{3}{4} = \frac{6}{4} + \frac{1}{4} - \frac{3}{4} = \frac{4}{4} = 1$$ Final: $$x^4 + \frac{1}{4} x^3 - \frac{3}{2} x^2 + \frac{3}{4} x$$ (xii) $a^2 b (a - b^2) + a b^2 (4 a b - 2 a^2) - a^3 b (1 - 2 b)$ Step 1: Expand: $$a^2 b \times a = a^3 b$$ $$a^2 b \times (- b^2) = - a^2 b^3$$ $$a b^2 \times 4 a b = 4 a^2 b^3$$ $$a b^2 \times (- 2 a^2) = - 2 a^3 b^2$$ $$- a^3 b \times 1 = - a^3 b$$ $$- a^3 b \times (- 2 b) = + 2 a^3 b^2$$ Step 2: Combine all: $$a^3 b - a^2 b^3 + 4 a^2 b^3 - 2 a^3 b^2 - a^3 b + 2 a^3 b^2$$ Step 3: Simplify: $$a^3 b - a^3 b = 0$$ $$- a^2 b^3 + 4 a^2 b^3 = 3 a^2 b^3$$ $$- 2 a^3 b^2 + 2 a^3 b^2 = 0$$ Answer: $3 a^2 b^3$ (xiii) $a^3 b (a^3 - a + 1) - a b (a^4 - 2 a^2 + 2 a) - b (a^3 - a^2 - 1)$ Step 1: Expand: $$a^3 b \times a^3 = a^6 b$$ $$a^3 b \times (- a) = - a^4 b$$ $$a^3 b \times 1 = a^3 b$$ $$- a b \times a^4 = - a^5 b$$ $$- a b \times (- 2 a^2) = 2 a^3 b$$ $$- a b \times 2 a = - 2 a^2 b$$ $$- b \times a^3 = - a^3 b$$ $$- b \times (- a^2) = a^2 b$$ $$- b \times (-1) = b$$ Step 2: Combine all: $$a^6 b - a^4 b + a^3 b - a^5 b + 2 a^3 b - 2 a^2 b - a^3 b + a^2 b + b$$ Step 3: Simplify like terms: $$a^3 b + 2 a^3 b - a^3 b = 2 a^3 b$$ $$- 2 a^2 b + a^2 b = - a^2 b$$ Final: $$a^6 b - a^5 b - a^4 b + 2 a^3 b - a^2 b + b$$ --- All problems solved with detailed steps and final answers.