Find each of the following product: \[\left( \frac{4}{3} u^2 vw \right) \times \left( - 5uv w^2 \right) \times \left( \frac{1}{3} v^2 wu \right)\]
Solution
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:
\[\left( \frac{4}{3} u^2 vw \right) \times \left( - 5uv w^2 \right) \times \left( \frac{1}{3} v^2 wu \right)\]
\[ = \left\{ \frac{4}{3} \times \left( - 5 \right) \times \frac{1}{3} \right\} \times \left( u^2 \times u \times u \right) \times \left( v \times v \times v^2 \right) \times \left( w \times w^2 \times w \right)\]
\[ = \left\{ \frac{4}{3} \times \left( - 5 \right) \times \frac{1}{3} \right\} \times \left( u^{2 + 1 + 1} \right) \times \left( v^{1 + 1 + 2} \right) \times \left( w^{1 + 2 + 1} \right)\]
\[ = - \frac{20}{9} u^4 v^4 w^4\]
Thus, the answer is \[- \frac{20}{9} u^4 v^4 w^4\].
