Question

How do you express $\dfrac{{{x}^{2}}}{{{y}^{-3}}}$ with positive exponents?

Answer 1

Hint: Here in this question, if powers are negative, we have to make them positive before reaching the final answer. You should be familiar with the properties of exponents and powers. We have to check whether bases are same or powers are same so that we apply a suitable identity for it. There is specifically the use of negative exponents in the given question.

Complete step by step answer:
Now, let’s solve the question.
As we know that exponents can be expressed in the form: ${{a}^{x}}$. They can be read as ‘a’ raise to the power ‘x’. Here, ‘a’ is the base and ‘x’ is the power. Remember that the value of ‘a’ should be greater than zero and cannot be equal to one. The value of ‘x’ can be any real number.
Let’s discuss some important functions for exponents. They are:
$\begin{align}
  & \Rightarrow {{a}^{x}}\times {{a}^{y}}={{a}^{x+y}} \\
 & \Rightarrow \dfrac{{{a}^{x}}}{{{a}^{y}}}={{a}^{x-y}} \\
 & \Rightarrow {{\left( {{a}^{x}} \right)}^{y}}={{a}^{xy}} \\
 & \Rightarrow {{a}^{x}}\times {{b}^{x}}={{\left( ab \right)}^{x}} \\
 & \Rightarrow \dfrac{{{a}^{x}}}{{{b}^{x}}}={{\left( \dfrac{a}{b} \right)}^{x}} \\
 & \Rightarrow {{a}^{0}}=1 \\
 & \Rightarrow {{a}^{-x}}=\dfrac{1}{{{a}^{x}}} \\
\end{align}$
Now, let’s talk about negative exponents. In this question specifically, we need to focus on negative exponents. When we talk about positive exponents, it gives us the result after being multiplied. But in negative exponents, we get the result after being divided. Let us see the difference with the help of some examples.
For positive exponent:
$\Rightarrow {{2}^{3}}=2\times 2\times 2$ which is equal to 8.
For negative exponent:
$\Rightarrow {{2}^{-3}}=\dfrac{1}{{{2}^{3}}}=\dfrac{1}{2\times 2\times 2}=\dfrac{1}{8}$ which is equal to 0.125
Now see, there is a huge difference between 8 and 0.125 similarly there is a huge difference between positive exponent and negative exponent.
Now in the question the expression is given:
$\Rightarrow \dfrac{{{x}^{2}}}{{{y}^{-3}}}$
Here we need to convert the negative power into positive one. We will get:
$\Rightarrow {{x}^{2}}{{y}^{3}}$
So this is our final answer.

Note:
In the final step, both the bases and powers are different, so we cannot apply any identity to solve the expression further. That’s why we have to leave the expression as it is. If a question comes to find a multiplicative inverse, then either to have 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 $\dfrac{1}{{{4}^{-9}}}$