Chapter 7.1, Problem 63PPS

a.

To determine

To complete: The given incomplete table using a calculator.

Answer to Problem 63PPS

The table is completed as shown.

  $\begin{array}{llllllllll}Power\hfill & 3^4\hfill & 3^3\hfill & 3^2\hfill & 3^1\hfill & 3^0\hfill & 3^{-1}\hfill & 3^{-2}\hfill & 3^{-3}\hfill & 3^{-4}\hfill \\ Value\hfill & 81\hfill & 27\hfill & 9\hfill & 3\hfill & 1\hfill & \frac{1}{3}\hfill & \frac{1}{9}\hfill & \frac{1}{27}\hfill & \frac{1}{81}\hfill \end{array}$

Explanation of Solution

Given Information: An incomplete table.

Calculation:

  $\begin{array}{llllllllll}Power\hfill & 3^4\hfill & 3^3\hfill & 3^2\hfill & 3^1\hfill & 3^0\hfill & 3^{-1}\hfill & 3^{-2}\hfill & 3^{-3}\hfill & 3^{-4}\hfill \\ Value\hfill & 81\hfill & 27\hfill & 9\hfill & 3\hfill & 1\hfill & \frac{1}{3}\hfill & \frac{1}{9}\hfill & \frac{1}{27}\hfill & \frac{1}{81}\hfill \end{array}$

Thus, table is completed.

b.

To determine

To find: The values of $5^0$ and $5^{-1}$ .

Answer to Problem 63PPS

The values are $1$ and $0.2$ respectively.

Explanation of Solution

Given Information: Values of $5^0$ and $5^{-1}$ are to be written.

Calculation:

  $5^0=1$ (Any number raised to the power $0$ is $1$ ).

  $\begin{array}{l}{5}^{-1}=\frac{1}{5}\\ \\ =0.2\end{array}$

Thus, values are $1$ and $0.2$ respectively.

c.

To determine

To complete: The given sentence.

Answer to Problem 63PPS

  $a^{-n}=\frac{1}{a^n}$ .

Explanation of Solution

Given Information: Sentence in regarding determining $a^{-n}$ .

Calculation:

For any nonzero number $a$ and any integer $n$ , $a^{-n}=\frac{1}{a^n}$ .

d.

To determine

To describe: The value of a non-zero number raised to the zero power.

Answer to Problem 63PPS

Value of any nonzero number raised to the zero power is $1$ .

Explanation of Solution

Given Information: Number is raised to the zero power.

Calculation:

Let nonzero number be $a$ ,

A number divided by itself is $1$ , i.e. $\frac{a^n}{a^n}=1$ , it can be re-written as $a^n{a}^{-n}=1$ , further simplifying by multiplying, $a^{n+(-n)}=1$ , this gives the result $a^0=1$ , this implies that value of any nonzero number raised to the zero power is $1$ .

Thus, value of any nonzero number raised to the zero power is $1$ .