Question

Find each of the following product: \[\left( \frac{7}{9}a b^2 \right) \times \left( \frac{15}{7}a c^2 b \right) \times \left( - \frac{3}{5} a^2 c \right)\]

Answer 1

To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., \[a^m \times a^n = a^{m + n}\].

We have: 

\[\left( \frac{7}{9}a b^2 \right) \times \left( \frac{15}{7}a c^2 b \right) \times \left( - \frac{3}{5} a^2 c \right)\]

\[ = \left\{ \frac{7}{9} \times \frac{15}{7} \times \left( - \frac{3}{5} \right) \right\} \times \left( a \times a \times a^2 \right) \times \left( b^2 \times b \right) \times \left( c^2 \times c \right)\]

\[ = \left\{ \frac{7^1}{9_3} \times \frac{{15}^3}{7} \times \left( - \frac{3^1}{5} \right) \right\} \times \left( a \times a \times a^2 \right) \times \left( b^2 \times b \right) \times \left( c^2 \times c \right)\]

\[ = \left\{ \frac{7^1}{9_3} \times \frac{{15}^{3^1}}{7} \times \left( - \frac{3^1}{5} \right) \right\} \times \left( a^{1 + 1 + 2} \right) \times \left( b^{2 + 1} \right) \times \left( c^{2 + 1} \right)\]

\[ = - a^4 b^3 c^3\]

Thus, the answer is \[- a^4 b^3 c^3\].