Express each of the following product as a monomials and verify the result for x = 1, y = 2:

Solution:

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:

$\left(-x y^{3}\right) \times\left(y x^{3}\right) \times(x y)$

$=(-1) \times\left(x \times x^{3} \times x\right) \times\left(y^{3} \times y \times y\right)$

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

$=-x^{5} y^{5}$

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

LHS $=\left(-x y^{3}\right) \times\left(y x^{3}\right) \times(x y)$

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

$=\{(-1) \times 1 \times 8\} \times(2 \times 1) \times 2$

$=(-8) \times 2 \times 2$

$=-32$

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

$\mathrm{RHS}=-x^{5} y^{5}$

$=(-1)(1)^{5}(2)^{5}$

$=(-1) \times 1 \times 32$

$=-32$

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

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