How do you simplify \[\dfrac{{2{x^{ - \dfrac{7}{4}}}}}{{4{x^{\dfrac{4}{3}}}}}\]?

Hint: Here, we are given the fraction which contains the same variable with different power. Therefore, we need to use the rules of exponent to simplify this term. We will be also using the concept of addition of fraction to get our final answer.

Complete step by step answer:
We are given \[\dfrac{{2{x^{ - \dfrac{7}{4}}}}}{{4{x^{\dfrac{4}{3}}}}}\].
First, we will use the negative exponent rule ${k^{ - m}} = \dfrac{1}{{{k^m}}}$ and move \[{x^{ - \dfrac{7}{4}}}\] to the denominator.
\[ \Rightarrow \dfrac{2}{{4{x^{\dfrac{4}{3}}}{x^{\dfrac{7}{4}}}}}\]
Now, we will multiply \[{x^{\dfrac{4}{3}}}\] and \[{x^{\dfrac{7}{4}}}\]. For this we will use the power rule ${k^m}{k^n} = {k^{m + n}}$ to combine exponents.
\[ \Rightarrow \dfrac{2}{{4{x^{\dfrac{4}{3} + \dfrac{7}{2}}}}}\]
Now, we will need to add two fractions \[\dfrac{4}{3} + \dfrac{7}{4}\].
Now. We can see that the denominators of both the terms are different. Therefore, we need to use the LCM method to solve it.
We know that the LCM of 3 and 4 is 12. Therefore, we can write:
\[\dfrac{4}{3} + \dfrac{7}{4} = \dfrac{{4 \times 4}}{{12}} + \dfrac{{7 \times 3}}{{12}} = \dfrac{{16 + 21}}{{12}} = \dfrac{{37}}{{12}}\]
\[ \Rightarrow \dfrac{2}{{4{x^{\dfrac{{37}}{{12}}}}}}\]
We have 4 in the denominator and we will now write it in the factor form as $4 = 2 \times 2$.
\[ \Rightarrow \dfrac{2}{{\left( {2 \times 2} \right){x^{\dfrac{{37}}{{12}}}}}}\]
We know that when there are similar terms in numerator and denominator, they will get cancelled out. Therefore, here 2 will get cancelled out and we will get
\[ \Rightarrow \dfrac{1}{{2{x^{\dfrac{{37}}{{12}}}}}}\]
Thus, by simplifying \[\dfrac{{2{x^{ - \dfrac{7}{4}}}}}{{4{x^{\dfrac{4}{3}}}}}\], we get \[\dfrac{1}{{2{x^{\dfrac{{37}}{{12}}}}}}\] as our final answer.

Note: While solving this question we have used the following rules:
The rule of negative exponent.
The addition of the exponent rule while multiplying two terms.
The rule of addition of fraction when denominators are not the same by using LCM.