Solution

\[ \begin{align*} & = (9 \times 2 ) \times (x^3 \times x^4) \\ & = 18 \times x^7 \\ & = 18x^7 \end{align*} \]

Answer \( \color{red} 18x^7 \)

Solution

\[ \begin{align*} & = (-6 \times 5) \times (a^2 \times a^7) \\ & = -30 \times a^{9} \\ & = -30a^9 \end{align*} \]

Answer \( \color{red} -30a^9 \)

Solution

\[ \begin{align*} & = (-8 \times -4) \times (y^9 \times y^3) \\ & = 32 \times y^{12} \\ & = 32y^{12} \end{align*} \]

Answer \( \color{red} 32y^{12} \)

Solution

\[ \begin{align*} & = 7pq \times \frac{4}{3} p^2 q^3 \\ \\ & = \left(7 \times \frac{4}{3} \right) \times (p^1 \times p^2) \times (q^1 \times q^3) \\ \\ & = \frac{28}{3} \times p^3 \times q^4 \\ \\ & = \frac{28}{3} p^3 q^4 \end{align*} \]

Answer \( \color{red} \frac{28}{3} p^3 q^4 \)

Solution

\[ \begin{align*} & = 12a^2b^6c^8 \times (-3a^7b^4c^3) \\ \\ & = [12 \times (-3)] \times (a^2 \times a^7) \times (b^6 \times b^4) \times (c^8 \times c^3) \\ \\ & = -36 \times a^9 \times b^{10} \times c^{11} \\ \\ & = -36a^9b^{10}c^{11} \end{align*} \]

Answer \( \color{red} -36a^9b^{10}c^{11} \)

Solution

\[ \begin{align*} & = \frac{2}{5} x^2 y \times \frac{5}{3} x^3 y^2 z^2 \\ \\ & = \left(\frac{2}{\cancel5} \times \frac{\cancel5}{3}\right) \times (x^2 \times x^3) \times (y^1 \times y^2) \times z^2 \\ \\ & = \frac{2}{3} \times x^5 \times y^3 \times z^2 \\ \\ & = \frac{2}{3} x^5 y^3 z^2 \end{align*} \]

Answer \( \color{red} \frac{2}{3} x^5 y^3 z^2 \)

Solution

\[ \begin{align*} & = 3x^7 \times 4x^2 \times (-5x^3) \\ & = (3 \times 4 \times -5) \times (x^7 \times x^2 \times x^3) \\ & = -60 \times x^{12} \\ & = -60x^{12} \end{align*} \]

Answer \( \color{red} -60x^{12} \)

Solution

\[ \begin{align*} & = 1.2a^2b^2 \times 5ab^4c^2 \times 1.1a^5bc^7 \\ & = (1.2 \times 5 \times 1.1) \times (a^2 \times a^1 \times a^5) \times (b^2 \times b^4 \times b^1) \times (c^2 \times c^7) \\ & = 6.6 \times a^8 \times b^7 \times c^9 \\ & = 6.6 a^8 b^7 c^9 \end{align*} \]

Answer \( \color{red} 6.6a^8b^7c^9 \)

Solution

\[ \begin{align*} & = \frac{3}{4}pq \times \frac{1}{2}qr^2 \times (-5p^2r^3) \times (-6r^5) \\ \\ & = \left[\frac{3}{4} \times \frac{1}{2} \times (-5) \times (-6) \right] \times (p^1 \times p^2) \times (q^1 \times q^1) \times (r^2 \times r^3 \times r^5) \\ \\ & = \left[ \frac{3}{4} \times \frac{1}{\cancel2} \times \cancel{30}^{15} \right]\times p^3 \times q^2 \times r^{10} \\ \\ & = \frac{45}{4}p^3q^2r^{10} \\ \end{align*} \]

Answer \( \color{red} \frac{45}{4}p^3q^2r^{10} \)

Answer

\[ \begin{align*} & = \left( \frac{1}{2} x^3 \right) \times (-10x) \times \left( \frac{1}{5} x^2 \right) \\ \\ & = \left[ \frac{1}{2} \times (-10) \times \frac{1}{5} \right] \times (x^3 \times x^1 \times x^2) \\ \\ & = \left( \frac{-10}{10} \right) \times x^{6} \\ \\ & = -1 \times x^6 \\ & = -x^6 \\ \\ & \color{magenta} \text{Verification } \\ \\ \color{magenta} LHS & = \color{green} \left( \frac{1}{2} x^3 \right) \times (-10x) \times \left( \frac{1}{5} x^2 \right) \\ \\ & = \left[\frac{1}{2} \times (1)^3 \right] \times (-10 \times 1) \times \left[ \frac{1}{5} \times (1)^2 \right] \\ \\ & = \left[\frac{1}{2} \times 1 \right] \times (-10 ) \times \left[ \frac{1}{5} \times 1 \right] \\ \\ & =\frac{1}{2} \times (-10 ) \times \frac{1}{5} \\ \\ & = \frac{-10}{10} \\ \\ & = -1 \\ \\ \color{magenta} RHS & = \color{green} -x^6 \\ & = -(1)^6 \\ & = -1 \\ \\ LHS & = RHS \\ \text{Hence } & \text{verified}\end{align*} \]

Solution

\[ \begin{align*} & = (-3xyz) \times \left( \frac{4}{9} x^2 z \right) \times \left( -\frac{27}{2} x y^2 z \right) \\ \\ & = \left[(-3) \times \frac{\cancel4^2}{\cancel9_1} \times \left( -\frac{\cancel{27}^3}{\cancel2_1} \right) \right] \times (x^1 \times x^2 \times x^1) \times (y^1 \times y^2) \times (z^1 \times z^1 \times z^1) \\ \\ & = \left[(-3) \times 2 \times ( -3) \right] \times x^4 \times y^3 \times z^3 \\ & = 18 \times x^4 \times y^3 \times z^3 \\ & = 18x^4 y^3 z^3 \\ \\ & { \color{magenta} \text{Verification: }} x = 2, y = 3, z = -1 \\ \\\color{magenta} LHS & = (-3xyz) \times \left( \frac{4}{9} x^2 z \right) \times \left( -\frac{27}{2} x y^2 z \right) \\ \\ & = [(-3) \times 2 \times 3 \times (-1)] \times \left( \frac{4}{9} \times 2^2 \times (-1) \right) \times \left( -\frac{27}{\cancel2} \times \cancel2 \times 3^2 \times (-1) \right) \\ \\ & = 18 \times \left( \frac{4 }{9} \times 4 \times (-1) \right) \times \left[ -27 \times 9 \times (-1) \right] \\ \\ & = \cancel{18}^2 \times \left( \frac{4 }{\cancel9} \times (-4) \right) \times \left( -27 \times -9 \right) \\ \\ & = 2 \times \left( -16 \right) \times 243 \\ & = (-32) \times 243 \\ & = -7776 \\ \\\color{magenta} RHS & = 18x^4 y^3 z^3 \\ & = 18 \times 2^4 \times 3^3 \times (-1)^3 \\ & = 18 \times 16 \times 27 \times (-1) \\ & = 7776 \times (-1) \\ & = -7776 \\ \\ LHS & = RHS \\ \text{Hence } & \text{verified} \end{align*} \]

Solution

\[ \begin{align*} & = (a^2bc^2) \times (9ab^2c^2) \times (-4ab^2c^4) \\ \\ & = \left[ 1 \times 9 \times (-4) \right] \times (a^2 \times a^1 \times a^1) \times (b^1 \times b^2 \times b^2) \times (c^2 \times c^2 \times c^4) \\ \\ & = (-36) \times a^{4} \times b^{5} \times c^{8} \\ \\ & = -36a^4b^5c^8 \\ \\ & {\color{magenta} \text{Verification: }} a = \frac{1}{2}, b = -1, c = 1 \\ \\\color{magenta} LHS & = (a^2bc^2) \times (9ab^2c^2) \times (-4ab^2c^4) \\ \\ & = \left( \left( \frac{1}{2} \right)^2 \times (-1) \times 1^2 \right) \times \left( 9 \times \frac{1}{2} \times (-1)^2 \times 1^2 \right) \times \left( -4 \times \frac{1}{2} \times (-1)^2 \times 1^4 \right) \\ \\ & = \left( \frac{1}{4} \times (-1) \times 1 \right) \times \left( 9 \times \frac{1}{2} \times 1 \times 1 \right) \times \left( -{\cancel4^2} \times \frac{1}{\cancel2} \times 1 \times 1 \right) \\ \\ & = \left( -\frac{1}{4} \right) \times \frac{9}{\cancel2_1} \times \left( -\cancel2^1 \right) \\ \\ & = \left( -\frac{1}{4} \right) \times 9 \times \left( -1 \right) \\ \\ & = \frac{9}{4} \\ \\ \color{magenta} RHS & = -36a^4b^5c^8 \\ & = -36 \times \left( \frac{1}{2} \right)^4 \times (-1)^5 \times (1)^8 \\ \\ & = -36 \times \frac{1}{16} \times (-1) \times 1 \\ \\ & = \frac{\cancel{36}^9}{\cancel{16}_4} \\ \\ & = \frac{9}{4} \\ \\ LHS & = RHS \\ \text{Hence } & \text{verified} \end{align*} \]

Solution

\[ \begin{align*} Let \\ \text{Length} & = 3a \\ \text{Breadth} & = 2a \\ \\ \text{Area of rectangle} & = \text{Length} \times \text{Breadth} \\ & = 3a \times 2a \\ & = 6a^2 \text{ square units} \end{align*} \]

Answer Area \( = \color{red} 6a^2 \text{ square units} \)

Solution

\[ \begin{align*} \text{Breadth} & = 4x \\ \text{Length} & = 3 \times \text{Breadth} \\ & = 3 \times 4x \\ & = 12x \\ \\ \text{Area of rectangle} & = \text{Length} \times \text{Breadth} \\ & = 12x \times 4x \\ & = 48x^2 \text{ square units} \end{align*} \]

Answer Area \( = \color{red} 48x^2 \text{ square units}\)

Solution

\[ \begin{align*} \text{Breadth} & = b \\ \text{Length} & = b^2 \\ \\ \text{Area of rectangle} & = \text{Length} \times \text{Breadth} \\ & = b^2 \times b \\ & = b^3 \text{ square units} \end{align*} \]

Answer Area \( = \color{red} b^3 \text{ square units} \)