Question

MULTIPLE REPRESENTATIONS in this problem, you will explore exponents.

a. TABULAR Copy and use a calculator to complete the table.

b. ANALYTICAL What do you think the values of${\mathbf{5}}^0$ and${\mathbf{5}}^{-\mathbf{1}}$ are? Verify your conjecture using a calculator.

c. ANALYTICAL Complete: for any nonzero number a and any integer n,$a^{-n}=\_.$

d. VERBAL Describe the value of a nonzero number raised to the zero power.

Short answer

a.

Power	$3^4$	$3^3$	$3^2$	$3^1$	$3^0$	$3^{-1}$	$3^{-2}$	$3^{-3}$	$3^{-4}$
Value	81	27	9	3	1	$\frac{1}{3}$	$\frac{1}{9}$	$\frac{1}{27}$	$\frac{1}{81}$

b. The values of$5^0$ and$5^{-1}$ are 1 and $\frac{1}{5}$.

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

d. For any nonzero number raised to the zero power is 1.

Step-by-step solution

Part a. Step 1. Find the values using calculator.

Enter a number then press exponent key and then enter a number in power.

The values in the table are

$\begin{array}{c}{3}^0=1\\ 3^1=3\\ 3^2=9\\ 3^3=27\\ 3^4=81\end{array}$

Part a. Step 2. Make a table.

Complete the table.

Power	$3^4$	$3^3$	$3^2$	$3^1$	$3^0$	$3^{-1}$	$3^{-2}$	$3^{-3}$	$3^{-4}$
Value	81	27	9	3	1	$\frac{1}{3}$	$\frac{1}{9}$	$\frac{1}{27}$	$\frac{1}{81}$

Part b. Step 1. Observe the table.

From the table in a., it can be observed that by using calculator the values of powers of 3 are

$\begin{array}{c}{3}^0=1\\ 3^{-1}=\frac{1}{3}\end{array}$

Part b. Step 2. Write the conclusion.

If $3^0=1$, then$5^0$ may also equal 1. If $3^{-1}=\frac{1}{3}$, then$5^{-1}$ may equal to $\frac{1}{5}$.

Part c. Step 1. Observe the table.

From the table, it can be observed that a number raised to a negative power is same as its reciprocal raised to the same power, that is,

$\begin{array}{c}{3}^{-1}=\frac{1}{3}\\ 3^{-2}=\frac{1}{3^2}\end{array}$

Part c. Step 2. Write the conclusion.

For any nonzero number a and any integer n,

$$\begin{gathered} a^{-n}={\left(\frac{1}{a}\right)}^n \\ =\frac{1}{a^n} \end{gathered}$$

Part d. Step 1. Observation.

From a. and b., it can be observed that for nonzero numbers, 3 and 5,$3^0=1$ and $5^0=1$.

Part d. Step 2. Write the conclusion.

From the above observation, it can be concluded that for any nonzero number x raised to the zero power is 1, that is, $x^0=1$.