How do you simplify $\dfrac{{4{x^3}}}{{2x}}$ .

Hint: To simplify this type of function at first we have to look at the numerator and denominator. If there will be any conjugate terms in the numerator and denominator at first we have to factorize them and write them as the multiplication of the factors. The common terms will vanish or cut out from the main function and the function will be simplified.

Formula used: $\dfrac{{a \times b}}{{a \times c}} = \dfrac{b}{c}$

Complete step by step solution:
We have;
 $\dfrac{{4{x^3}}}{{2x}}$
At first, we will factorize the terms in the numerator and the denominator.
The numerator $4{x^3}$ is the multiplication of $5$ terms and they are $2$ , $2$ , $x$ , $x$ , $x$ and $2\;x$ is the multiplication of $2$ and $x$ . Then we can write;
$ = \dfrac{{2 \times 2 \times x \times x \times x}}{{2 \times x}}$
We can see that the terms $2$ and $x$ are common in the numerator and the denominator so we will separate them from the other terms in the numerator and the denominator ;
$ = \dfrac{{(2 \times x) \times (2 \times x \times x)}}{{(2 \times x) \times (1)}}$
We know that $\dfrac{{a \times b}}{{a \times c}} = \dfrac{b}{c}$ .
By this formula we will measure $2 \times x$ as a single term and $2 \times x \times x$ as a single term then we can simplify.
Then clearly the common part $2 \times x$ will be cut out or vanished and we will have;
$ = \dfrac{{2 \times x \times x}}{1}$
Multiplying all these terms of numerator and denominator we get;
$ = 2{x^2}$

So, the simplified form is $2{x^2}$ and this is the required solution.

Note: Students always observe the more simplified or less factorable term among the numerator and denominator. The numerator may be less factorable than the denominator, then you have to find the factors of the numerator and then simplify. Factorization will make the problem easy and then the problem will be simplified.