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.

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

The values in the table are

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

Complete the table.

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

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

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

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,

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

For any nonzero number a and any integer n,

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

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

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