Question

How do you simplify $\dfrac{{{x}^{4}}}{{{x}^{9}}}$?

Answer 1

Hint: Here in this question, we have given the same base in numerator and denominator. So the powers will be subtracted by using quotient rule: $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$. And also solutions will be expressed in positive exponents only and if powers become negative, we have to make them positive before reaching the final answer. You should be familiar with the properties of exponents and powers.

Complete step by step answer:
Now, let’s solve the question.
As we already know that exponents can be expressed in the form: ${{a}^{x}}$ and can be read as ‘a’ raise to the power ‘x’. Where ‘a’ is the base and ‘x’ is the power. Keep in mind that the value of ‘a’ should be greater than zero and cannot be equal to one. The value of ‘x’ can be a real number.
Let’s discuss some important functions for exponents.
$\begin{align}
  & \Rightarrow {{a}^{x}}\times {{a}^{y}}={{a}^{x+y}} \\
 & \Rightarrow \dfrac{{{a}^{x}}}{{{a}^{y}}}={{a}^{x-y}} \\
 & \Rightarrow {{\left( {{a}^{x}} \right)}^{y}}={{a}^{xy}} \\
 & \Rightarrow {{a}^{x}}\times {{b}^{x}}={{\left( ab \right)}^{x}} \\
 & \Rightarrow \dfrac{{{a}^{x}}}{{{b}^{x}}}={{\left( \dfrac{a}{b} \right)}^{x}} \\
 & \Rightarrow {{a}^{0}}=1 \\
 & \Rightarrow {{a}^{-x}}=\dfrac{1}{{{a}^{x}}} \\
\end{align}$
Now, write the expression given in question:
$\Rightarrow \dfrac{{{x}^{4}}}{{{x}^{9}}}$
First step is to identify which property is best suitable in solving the above expression. And we can see that the bases are the same i.e. ‘x’ and powers are different but they are dividing each other. So we will apply quotient rule:
 $\Rightarrow \dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
Apply the rule now:
$\Rightarrow {{x}^{4-9}}={{x}^{-5}}$
Now we can see that we got the negative power. We have to make it positive by applying multiplicative inverse:
$\Rightarrow \dfrac{1}{{{x}^{5}}}$
This is our final answer.

Note: In multiplicative inverse, we have to make the power negative if it is given positive or make positive if it is given as negative. For example:
$\Rightarrow $Multiplicative inverse of ${{2}^{-5}}$ is equal to $\dfrac{1}{{{2}^{5}}}$
$\Rightarrow $Multiplicative inverse of ${{4}^{9}}$ is equal to $\dfrac{1}{{{4}^{-9}}}$
Remember all the rules of exponents and powers. Keep in mind that the final answer cannot be in a negative power. If it is, then make it a positive one.