Question

How do you simplify \[\dfrac{x}{{{x^3}}}?\]

Answer 1

Hint: This question involves the arithmetic operation like addition/ subtraction/ multiplication/ division. Also, we need to know basic algebraic formulae with the involvement of fraction terms. Also, we need to know how to convert negative power terms into positive power terms. We need to know the algebraic formulae related to exponent components.

Complete step-by-step answer:
The given problem is shown below,
 \[\dfrac{x}{{{x^3}}} = ?\]
The above equation can also be written as,
 \[\dfrac{{{x^1}}}{{{x^3}}} = ? \to \left( 1 \right)\]
We know that,
 \[\dfrac{{{x^a}}}{{{x^b}}} = {x^{a - b}} \to \left( 2 \right)\]
For solving the given problem, let’s substitute the equation \[\left( 2 \right)\] in the equation \[\left( 1 \right)\] , we get
 \[
  \left( 1 \right) \to \dfrac{{{x^1}}}{{{x^3}}} = ? \\
  \dfrac{{{x^1}}}{{{x^3}}} = {x^{1 - 3}} \\
 \] (Here \[a = 1\] and \[b = 3\] )
We know that, \[1 - 3 = - 2\]
So, we get
 \[\dfrac{{{x^1}}}{{{x^3}}} = {x^{ - 2}} \to \left( 2 \right)\]
We know that,
 \[\dfrac{1}{{{x^a}}} = {x^{ - a}} \to \left( 3 \right)\]
By using the equation \[\left( 3 \right)\] in the equation \[\left( 2 \right)\] , we get
 \[\dfrac{{{x^1}}}{{{x^3}}} = \dfrac{1}{{{x^2}}}\] (Here \[a = 2\] )
So, the final answer is,
 \[\dfrac{x}{{{x^3}}} = \dfrac{1}{{{x^2}}}\]
So, the correct answer is “ \[\dfrac{x}{{{x^3}}} = \dfrac{1}{{{x^2}}}\] ”.

Note: In this type of question we would involve the arithmetic operation of addition/ subtraction/ multiplication/ division. Note that if there is no term mentioned in power \[\left( x \right)\] , we can mention the power term as \[1\left( {{x^1}} \right)\] . Also, note that if a negative power term is placed in the numerator, it converts into a positive power term when we move it to the denominator place \[\left( {{x^{ - n}} \to \dfrac{1}{{{x^n}}}} \right)\] . The given problem can also be solved by finding the greatest common factor between the numerator and the denominator. In the given question we have \[x\] as a common factor between the numerator and the denominator. So, by dividing the numerator term and denominator term with \[x\] we get the final answer is \[\dfrac{1}{{{x^2}}}\] . By using the above-mentioned method we can simplify the given question within two or three steps.