Question

How do you express ${{d}^{-3}}$ as a positive exponent?

Answer 1

Hint: In this question we have an exponential term which has a negative power therefore, we will use the law of exponents and state what a negative exponent is by using an example and use that property on the exponent to write the exponent as a positive exponent. We will derive how to convert a negative exponent into a positive exponent.

Complete step-by-step solution:
We have the term given as ${{d}^{-3}}$, which we have to convert from a negative exponent to a positive exponent.
To understand what a negative exponent represents we first need to understand how a positive exponent works.
Consider a term ${{a}^{b}}$, the term is in the exponential form which means that the value of the term will be $a$ multiplied by itself for $b$ number of times. For example, if the value of $a=2$ and $b=3$ then ${{a}^{b}}={{2}^{3}}$
Which can be simplified as ${{a}^{b}}=2\times 2\times 2=8$, which is its value.
Now suppose you want the value of ${{2}^{2}}$, so it can be calculated by dividing ${{2}^{3}}$ by $2$.
It can be written as:
$\Rightarrow {{2}^{2}}=\dfrac{{{2}^{3}}}{2}$
This implies that when the value of ${{a}^{b}}=k$ then the value of ${{a}^{c}}$ where $c< b$, we can find the value by using the general formula ${{a}^{b-c}}$
Now we know that ${{n}^{0}}=1$ so ${{n}^{-t}}$ must be divided by $\left( 0-\left( -t \right) \right)$ times which means:
$\Rightarrow {{n}^{-t}}=\dfrac{1}{{{n}^{t}}}$
Now using this equation on the term ${{d}^{-3}}$, we get:
$\Rightarrow {{d}^{-3}}=\dfrac{1}{{{d}^{3}}}$, which is the required answer.

Note: The general property of exponents should be remembered while doing these types of sums. It is also to be remembered when the power to a term is $0$ then the solution will always be $1$. It is not to be mistaken as ${{n}^{0}}=0$. The above property is also called as reciprocal which means dividing the term by $1$ such that the product of a term and its reciprocal is $1$.