Answer

\[ \begin{align*} & = (x^2 + 3xy) \times 4x \\ & = (x^2 \times 4x) + (3xy \times 4x) \\ & = 4x^3 + 12x^2y \end{align*} \]

Answer

\[ \begin{align*} & = 9pqr \times (2p^2q - 3q^2r) \\ & = (9pqr \times 2p^2q) - (9pqr \times 3q^2r) \\ & = 18p^3q^2r - 27pq^3r^2 \end{align*} \]

Answer

\[ \begin{align*} & = \frac{3}{4}a^2 \times \left( \frac{2}{3}b^2 + 8ab \right) \\ \\ & = \left( \frac{\cancel3}{\cancel4_2}a^2 \times \frac{\cancel2^1}{\cancel3}b^2 \right) + \left( \frac{3}{\cancel4_1}a^2 \times {\cancel8^2}ab \right) \\ \\ & = \left( \frac{1}{2}a^2 \times b^2 \right) + \left( 3a^2 \times 2ab \right) \\ \\ & = \frac{1}{2}a^2b^2 + 6a^3b \end{align*} \]

Solution

\[ \begin{align*} & = (7x + 9y^2) \times 3xy^2 \\ & = (7x \times 3xy^2) + (9y^2 \times 3xy^2) \\ & = \color{green} 21x^2y^2 + 27xy^4 \\ \\& \text{For } x = 2, y = -1 \\ \\ & = \left[21(2)^2(-1)^2 \right] + \left[27(2)(-1)^4 \right] \\ \\ & = \left[21 \times 4 \times 1 \right] + [27 \times 2 \times 1] \\ & = 84 + 54 \\ & = 138 \end{align*} \]

Answer \( \color{red} 21x^2y^2 + 27xy^4 \) and \( \color{red} 138 \)

Solution

\[ \begin{align*} & = -11x \times (2y^5 - \frac{3}{11}x^2y^3) \\ \\ & = (-11x \times 2y^5) - \left(-\cancel{11}x \times \frac{3}{\cancel{11}}x^2y^3\right) \\ \\ & = (-22xy^5) - \left(-3x^3y^3\right) \\ & = \color{green} -22xy^5 + 3x^3y^3 \\ \\ & \text{For } x = 2, y = -1 \\ \\ & = \left[-22(2)(-1)^5\right] + \left[3(2)^3(-1)^3\right] \\ \\ & = \left[-22 \times 2 \times -1 \right] + \left[3 \times 8 \times -1\right] \\ & = 44 - 24 \\ & = 20 \end{align*} \]

Answer \( \color{red} -22xy^5 + 3x^3y^3 \) and \( \color{red} 20 \)

Solution

\[ \begin{align*} & = 0.5x \times (2x^2y^2 + 1.5xy^3) \\ & = (0.5x \times 2x^2y^2) + (0.5x \times 1.5xy^3) \\ & = \color{green} 1x^3y^2 + 0.75x^2y^3 \\ \\ & \text{For } x = 2, y = -1 \\ \\ & = \left[1(2)^3(-1)^2\right] + \left[0.75(2)^2(-1)^3\right] \\ \\ & = \left[1 \times 8 \times 1 \right] + \left[0.75 \times 4 \times -1\right] \\ & = \left[8 \right] + [-3] \\ & = 8 - 3 \\ & = 5 \end{align*} \]

Answer \( \color{red} x^3y^2 + 0.75x^2y^3 \) and \( \color{red} 5 \)

Answer

\[ \begin{align*} & = 8s^2 \times (t^2 - 2st) \\ & = (8s^2 \times t^2) - (8s^2 \times 2st) \\ & = 8s^2t^2 - 16s^3t \\ \\ & \color{magenta} \text{Verification} \\ \\\color{magenta} LHS & = \color{green} 8s^2 \times (t^2 - 2st) \\ & = 8 (1)^2 \times [(5)^2 - 2(1)(5)] \\ & = 8 \times [25 - 10] \\ & = 8 \times 15 \\ & = 120 \\ \\\color{magenta} RHS & = \color{green} 8s^2t^2 - 16s^3t \\ & = 8 (1)^2 (5)^2 - 16 (1)^3 (5) \\ & = (8 \times 1 \times 25) - (16 \times 1 \times 5) \\ & = 200 - 80 \\ & = 120 \\ \\ LHS & = RHS \\ \text{Hence } & \text{verified.} \end{align*} \]

Answer

\[ \begin{align*} & = \frac{2}{7} x^2 \times (7y + 14x) \\ \\ & = \left(\frac{2}{\cancel7} x^2 \times \cancel7y\right) + \left(\frac{2}{\cancel7_{\color{blue}1}} x^2 \times \cancel{14}^{\color{blue}2}x\right) \\ \\ & = (2x^2 \times y) + (2x^2 \times 2x) \\ & = 2x^2y + 4x^3 \\ \\ & \color{magenta} \text{Verification} \\ \\ \color{magenta} LHS & = \color{green} \frac{2}{7} x^2 \times (7y + 14x) \\ \\ & = \frac{2}{7} (2)^2 \times [7(3) + 14(2)] \\ \\ & = \frac{2}{7} \times 4 \times [21 + 28] \\ \\ & = \frac{8}{\cancel7_{\color{blue}1}} \times \cancel{49}^{\color{blue}7} \\ \\ & = 8 \times 7 \\ & = 56 \\ \\ \color{magenta} RHS & = \color{green} 2x^2y + 4x^3 \\ & = 2(2)^2(3) + 4(2)^3 \\ & = (2 \times 4 \times 3) + (4 \times 8) \\ & = 24 + 32 \\ & = 56 \\ \\ LHS & = RHS \\ \text{Hence } & \text{verified.} \end{align*} \]

Answer

\[ \begin{align*} & = 0.2xy \times (3x + 2y) \\ & = \left(0.2xy \times 3x\right) + \left(0.2xy \times 2y\right) \\ & = 0.6x^2y + 0.4xy^2 \\ \\ & \color{magenta} \text{Verification} \\ \\ \color{magenta} LHS & = \color{green} 0.2xy \times (3x + 2y) \\ & = 0.2(5)(-1) \times [3(5) + 2(-1)] \\ & = -1 \times [15 - 2] \\ & = -1 \times 13 \\ & = -13 \\ \\ \color{magenta} RHS & = \color{green} 0.6x^2y + 0.4xy^2 \\ & = 0.6(5)^2(-1) + 0.4(5)(-1)^2 \\ & = [0.6 \times 25 \times (-1)] + [0.4 \times 5 \times 1] \\ & = [15 \times (-1)] + [2 \times 1] \\ & = -15 + 2 \\ & = -13 \\ \\ LHS & = RHS \\ \text{Hence } & \text{verified.} \end{align*} \]

Answer

\[ \begin{align*} & = ab(a^2 - b^2) + b^3(a - 2b) \\ & = (ab \times a^2) - (ab \times b^2) + (b^3 \times a) - (b^3 \times 2b) \\ & = a^3b - \cancel{ab^3} + \cancel{ab^3} - 2b^4 \\ & = a^3b - 2b^4 \end{align*} \]

Answer

\[ \begin{align*} & = -6x^2(xy + 2y^2) - 3y^2(2x^2 + y) \\ & = -6x^2 \times xy -6x^2 \times 2y^2 -3y^2 \times 2x^2 -3y^2 \times y \\ & = -6x^3y {\color{magenta} - 12x^2y^2- 6x^2y^2} - 3y^3 \\ & = -6x^3y - 18x^2y^2 - 3y^3 \end{align*} \]

Answer

\[ \begin{align*} & = 2x(x - y^2) - 3y(xy + 2x) - xy(x + y) \\ & = 2x \times x - 2x \times y^2 - 3y \times xy - 3y \times 2x - xy \times x - xy \times y \\ & = 2x^2 {\color{magenta} - 2xy^2 - 3xy^2} - 6xy - x^2y {\color{magenta} - xy^2} \\ & = 2x^2 - 6xy^2 - x^2y - 6xy \end{align*} \]