Question

Express each of the following product as a monomials and verify the result in each case for x = 1:
(5x⁴) × (x²)³ × (2x)²

Answer 1

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., $a^m\times a^n=a^{m+n}\text{and}{\left(a^m\right)}^n=a^{mn}$.

We have:

$$\begin{gathered} \left(5x^4\right)\times {\left(x^2\right)}^3\times {\left(2x\right)}^2 \\ =\left(5x^4\right)\times \left(x^6\right)\times \left(2^2\times x^2\right) \\ =\left(5\times 2^2\right)\times \left(x^4\times x^6\times x^2\right) \\ =\left(5\times 2^2\right)\times \left(x^{4+6+2}\right) \\ =20x^{12} \end{gathered}$$
$\therefore$ $\left(5x^4\right)\times {\left(x^2\right)}^3\times {\left(2x\right)}^2=20x^{12}$

Substituting x = 1 in LHS, we get:

$$\begin{gathered} \mathrm{LHS}=\left(5x^4\right)\times {\left(x^2\right)}^3\times {\left(2x\right)}^2 \\ =\left(5\times 1^4\right)\times {\left(1^2\right)}^3\times {\left(2\times 1\right)}^2 \\ =\left(5\times 1\right)\times \left(1^6\right)\times {\left(2\right)}^2 \\ =5\times 1\times 4 \\ =20 \end{gathered}$$

Put x =1 in RHS, we get:

$$\begin{gathered} \text{RHS}=20x^{12} \\ =20\times {\left(1\right)}^{12} \\ =20\times 1 \\ =20 \end{gathered}$$

$\because$ LHS = RHS for x = 1; therefore, the result is correct.

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