Chapter 7.1, Problem 63PPE

To determine

To find: The values for each $a^{-1}$ , $a^2$ and $a^{-2}$ by choosing a fraction to use a value for the given variable.

Answer to Problem 63PPE

The values of fraction $\frac{1}{5}$ for each $a^{-1}$ is $5$ , $a^2$ is $\frac{1}{25}$ and $a^{-2}$ is $25$ .

Explanation of Solution

Given information:

The given variable is $a$ .

Calculation:

Let the fraction for $a$ is:

  $\frac{1}{5}$

Formula for exponents raised to the negative power equals is:

  $a^{-n}=\frac{1}{a^n}$

Substitute $\frac{1}{5}$ for $a$ in the above expression.

  $\begin{array}{c}{\left(\frac{1}{5}\right)}^{-n}=\frac{1}{{\left(\frac{1}{5}\right)}^n}\\ ={\left(\frac{5}{1}\right)}^n\end{array}$

To calculate the value of $a^{-1}$ , substitute $-1$ for $n$ in the above expression.

  $\begin{array}{c}{\left(\frac{1}{5}\right)}^{-1}={\left(\frac{5}{1}\right)}^1\\ =\frac{5}{1}\\ =5\end{array}$

To calculate the value of $a^{-2}$ , substitute $-1$ for $n$ and $\frac{1}{5}$ for $a$ in the above expression.

  $\begin{array}{c}{\left(\frac{1}{5}\right)}^{-2}={\left(\frac{5}{1}\right)}^2\\ =\frac{5\times 5}{1\times 1}\\ =25\end{array}$

Formula for exponents raised to a positive power of $2$ is:

  $\begin{array}{c}{\left(\frac{1}{5}\right)}^2=\frac{1}{5}\times \frac{1}{5}\\ =\frac{1}{25}\end{array}$

Therefore, the values of $a^{-1}={\left(\frac{1}{5}\right)}^{-1}$ is $5$ , $a^2={\left(\frac{1}{5}\right)}^2$ is $\frac{1}{25}$ and $a^{-2}={\left(\frac{1}{5}\right)}^{-2}$ is $25$ .