Question

Question 54 what I’m suppose to do I’m not got is says evaluate

Answer 1

54. We need to find the value of the expression

$\frac{1}{12^{-1}}$

In order to do so, we can use the following definition for exponentials:

$\begin{gathered} x^{-1}=\frac{1}{x} \\ \\ \frac{1}{x^{-1}}=\frac{1}{\frac{1}{x^{}}}=1\cdot\frac{x}{1}=x \end{gathered}$

In this problem, we have x = 12. Thus, we can evaluate the expression as follows:

$\frac{1}{12^{-1}}=12$

Let's see another way to find the same result.

Notice that when a number has the positive exponent n, it means that we have a product of n factors of that number. For example:

$\begin{gathered} 2^3=2\cdot2\cdot2=8 \\ \\ 2^1=2 \end{gathered}$

On the other hand, when the exponent is negative, by definition, the minus sign tells us to find the inverse of that number. For example:

$\begin{gathered} 2^{-3}=\mleft(\frac{1}{2}\mright)^3=\mleft(\frac{1}{2}\mright)\cdot\mleft(\frac{1}{2}\mright)\cdot\mleft(\frac{1}{2}\mright)\text{ because }\frac{1}{2}\text{ is the inverse of }2 \\ \\ 2^{-1}=\frac{1}{2} \end{gathered}$

Also, notice that 1 to any exponent equals 1. For example:

$\begin{gathered} 1^0=1 \\ 1^1=1 \\ 1^2=1 \\ 1^{-1}=1 \end{gathered}$

So, we can write:

$\frac{1}{12^{-1}}=\frac{1^{-1}}{12^{-1}}=\mleft(\frac{1}{12}\mright)^{-1}=\mleft(\frac{12}{1}\mright)=12$

Therefore, the value of the given expression is 12.