Express each of the following product as a monomials and verify the result in each case for x = 1:

Solution:

We have to find the product of the expression in order to express it as a monomial.

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

We have:

$\left(5 x^{4}\right) \times\left(x^{2}\right)^{3} \times(2 x)^{2}$

$=\left(5 x^{4}\right) \times\left(x^{6}\right) \times\left(2^{2} \times x^{2}\right)$

$=\left(5 \times 2^{2}\right) \times\left(x^{4} \times x^{6} \times x^{2}\right)$

$=\left(5 \times 2^{2}\right) \times\left(x^{4+6+2}\right)$

$=20 x^{12}$

$\therefore\left(5 x^{4}\right) \times\left(x^{2}\right)^{3} \times(2 x)^{2}=20 x^{12}$

Substituting x = 1 in LHS, we get:

$\mathrm{LHS}=\left(5 x^{4}\right) \times\left(x^{2}\right)^{3} \times(2 x)^{2}$

$=\left(5 \times 1^{4}\right) \times\left(1^{2}\right)^{3} \times(2 \times 1)^{2}$

$=(5 \times 1) \times\left(1^{6}\right) \times(2)^{2}$

$=5 \times 1 \times 4$

$=20$

 

Put x =1 in RHS, we get:

RHS $=20 x^{12}$

$=20 \times(1)^{12}$

$=20 \times 1$

$=20$

$\because L H S=R H S$ for $x=1$; therefore, the result is correct.

Thus, the answer is $20 x^{12}$.