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:

\[\left( - x y^3 \right) \times \left( y x^3 \right) \times \left( xy \right)\]

\[ = \left( - 1 \right) \times \left( x \times x^3 \times x \right) \times \left( y^3 \times y \times y \right)\]

\[ = \left( - 1 \right) \times \left( x^{1 + 3 + 1} \right) \times \left( y^{3 + 1 + 1} \right)\]

\[ = - x^5 y^5\]

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

\[\text  { LHS }= \left( - x y^3 \right) \times \left( y x^3 \right) \times \left( xy \right)\]

\[ = \left\{ \left( - 1 \right) \times 1 \times 2^3 \right\} \times \left( 2 \times 1^3 \right) \times \left( 1 \times 2 \right)\]

\[ = \left\{ \left( - 1 \right) \times 1 \times 8 \right\} \times \left( 2 \times 1 \right) \times 2\]

\[ = \left( - 8 \right) \times 2 \times 2\]

\[ = - 32\]

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

\[\text { RHS } = - x^5 y^5 \]

\[ = \left( - 1 \right) \left( 1 \right)^5 \left( 2 \right)^5 \]

\[ = \left( - 1 \right) \times 1 \times 32\]

\[ = - 32\]

Because LHS is equal to RHS, the result is correct.
Thus, the answer is \[- x^5 y^5\].