Chapter P.1, Problem 65E

Solution

a.

To examine the equation $a^m{a}^n=a^{m+n}$ for $n=0$ and explain why it is reasonable to define $a^0=1\text{ for }a\ne 0$ .

Given:

The positive integers m and n .

Concept Used:

• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of powers property: $\frac{a^m}{a^n}=a^{m-n}$

Calculation:

In order to examine the equation $a^m{a}^n=a^{m+n}$ for $n=0$ , use the exponent properties and simplify further as shown below

  $\begin{array}{c}{a}^m\cdot a^0=a^{m+0}\\ =a^m\\ =a^m\cdot 1\end{array}$

It implies that $a^0=1\text{ for }a\ne 0$ .

b.

To examine the equation $a^m{a}^n=a^{m+n}$ for $n=-m$ and explain why it is reasonable to define $a^{-m}=\frac{1}{a^m}\text{ for }a\ne 0$ .

Given:

The positive integers m and n .

Concept Used:

• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of powers property: $\frac{a^m}{a^n}=a^{m-n}$

Calculation:

In order to examine the equation $a^m{a}^n=a^{m+n}$ for $n=-m$ , use the exponent properties and simplify further as shown below

  $\begin{array}{c}{a}^m\cdot a^n=a^m\cdot a^{-m}\\ =a^{m+\left(-m\right)}\\ =a^{m-m}\\ =a^0\\ =1\end{array}$

Since the product is 1, it implies that $a^{-m}=\frac{1}{a^m}\text{ for }a\ne 0$ .