Question

Express each of the following product as a monomials and verify the result for x = 1, y = 2:
(−xy³) × (yx³) × (xy)

Answer 1

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

We have:

$$\begin{gathered} \left(-x{y}^3\right)\times \left(y{x}^3\right)\times \left(xy\right) \\ =\left(-1\right)\times \left(x\times x^3\times x\right)\times \left(y^3\times y\times y\right) \\ =\left(-1\right)\times \left(x^{1+3+1}\right)\times \left(y^{3+1+1}\right) \\ =-x^5{y}^5 \end{gathered}$$

To verify the result, we substitute x = 1 and y = 2 in LHS; we get:

$$\begin{gathered} \text{LHS}=\left(-x{y}^3\right)\times \left(y{x}^3\right)\times \left(xy\right) \\ =\left\{\left(-1\right)\times 1\times 2^3\right\}\times \left(2\times 1^3\right)\times \left(1\times 2\right) \\ =\left\{\left(-1\right)\times 1\times 8\right\}\times \left(2\times 1\right)\times 2 \\ =\left(-8\right)\times 2\times 2 \\ =-32 \end{gathered}$$

Substituting x = 1 and y = 2 in RHS, we get:​

$$\begin{gathered} \text{RHS}=-x^5{y}^5 \\ =\left(-1\right){\left(1\right)}^5{\left(2\right)}^5 \\ =\left(-1\right)\times 1\times 32 \\ =-32 \end{gathered}$$

Because LHS is equal to RHS, the result is correct.

Thus, the answer is $-x^5{y}^5$.