Chapter 2.1, Problem 47E

Solution

To show: The negative exponent property can be derived from the quotient of powers property and the zero exponent property.

Given information:

Properties are the negative exponent property, quotient of powers property and the zero exponent property

Explanation:

The negative exponent property states that $\frac{1}{a^m}=a^{-m}$ .

Let us start with the left side of the equation and try deriving the expression on the right.

By the zero-exponent rule, $a^0=1$ . Replace $1$ with $a^0$ in $\frac{1}{a^m}$ .

  $\frac{1}{a^m}=\frac{a^0}{a^m}$

Now, the quotient of the two powers can be found by applying the quotient of powers property,

  $\begin{array}{c}\frac{a^m}{a^n}=a^{m-n}\\ \frac{a^0}{a^m}=a^{0-m}\\ =a^{-m}\end{array}$

The left side simplifies to the right side.

Thus, we have derived the negative exponent property the quotient of powers property and the zero exponent property.