How do you simplify \[\dfrac{{{x^{ - 1}}}}{{4{x^4}}}\] and write it using only positive exponents?

Hint:In this question we need to write this given expression \[\dfrac{{{x^{ - 1}}}}{{4{x^4}}}\] only using positive exponents. To solve this question we need to know the rules of exponents. If you know the rules then only you are able to solve .Rule of exponent which we use is${x^{ - 1}} = \dfrac{1}{x}$.

Complete step by step solution: Let us try to solve this in which we are asked to simplify expression \[\dfrac{{{x^{ - 1}}}}{{4{x^4}}}\] and write it using only positive exponents. To solve this type of question we need to know the law exponents.

Here are the two laws by using which we solve this question.

1) ${x^{ - 1}} = \dfrac{1}{x}$

2 ${x^a} \times {x^b} = {x^{a + b}}$

We will use these two laws of exponents to give solutions to solve these types of questions.
Using, ${x^{ - 1}} = \dfrac{1}{x}$

We have to write the expression \[\dfrac{{{x^{ - 1}}}}{{4{x^4}}}\] only using positive exponents.

 We know from the property $1$of exponents we can write our expression as
$\dfrac{{{x^{ - 1}}}}{{4{x^4}}} = \dfrac{1}{{4{x^4}}} \cdot \dfrac{1}{x}$

Now by using property $2$in our expression, we get
$
\dfrac{{{x^{ - 1}}}}{{4{x^4}}} = \dfrac{1}{{4{x^4}}} \cdot \dfrac{1}{x} \\
\,\,\,\,\,\,\,\,\,\, = \dfrac{1}{{4{x^{4 + 1}}}} \\
\,\,\,\,\,\,\,\,\, = \dfrac{1}{{4{x^5}}} \\
$
So expression \[\dfrac{{{x^{ - 1}}}}{{4{x^4}}}\] will be written as $\dfrac{1}{{4{x^5}}}$only using positive exponents.

Without using${x^{ - 1}} = \dfrac{1}{x}$,

We can write $\dfrac{{{x^{ - 1}}}}{{4{x^4}}} = \dfrac{{{x^{ - 1}}}}{{4{x^4}}} \cdot \dfrac{x}{x}$
$\dfrac{{{x^{ - 1}}}}{{4{x^4}}} = \dfrac{{{x^{ - 1}} \cdot x}}{{4{x^4} \cdot x}}$

Now using property$2$, we get
$
\dfrac{{{x^{ - 1}}}}{{4{x^4}}} = \dfrac{{{x^{ - 1 + 1}}}}{{4{x^{4 + 1}}}} \\
\,\,\,\,\,\,\,\,\,\, = \dfrac{{{x^0}}}{{4{x^5}}} \\
$
As we know that power of anything to zero is$1$.So${x^0} = 1$,
$\dfrac{{{x^{ - 1}}}}{{4{x^4}}} = \dfrac{1}{{4{x^5}}}$

So expression \[\dfrac{{{x^{ - 1}}}}{{4{x^4}}}\] will be written as $\dfrac{1}{{4{x^5}}}$only using positive exponents.

Note: These kinds of questions are very easy, we just need to know the rules of exponents. In this type of question students generally make mistakes in writing the sign of exponents, so be careful. Similarly we can also be asked to write expressions only using negative exponents.