Chapter 7.4, Problem 87PPE

(a).

To determine

To write: The expression $\frac{a^0}{a^n}$ as a power of $a$ by using property for dividing powers.

Answer to Problem 87PPE

The expression $\frac{a^0}{a^n}$ as a power of $a$ can be written as $a^{-n}$ .

Explanation of Solution

Given information: The given expression is $\frac{a^0}{a^n}$ and the property to be used is dividing powers with the same base.

Calculation:

The property of dividing powers with the same base for any positive integers $x,y$ is:

  $\frac{a^x}{a^y}=a^{x-y};a\ne 0$

Apply the property of dividing powers to simplify the expression $\frac{a^0}{a^n}$ .

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

Therefore, the expression $\frac{a^0}{a^n}$ as a power of $a$ can be written as $a^{-n}$ .

(b).

To determine

To simplify: The expression $\frac{a^0}{a^n}$ by using a zero exponent.

Answer to Problem 87PPE

The simplified form of expression $\frac{a^0}{a^n}$ by using a zero exponent is $\frac{1}{a^n}$ .

Explanation of Solution

Given information: The given expression is $\frac{a^0}{a^n}$ and the property to be used is zero exponent.

Calculation:

The zero exponent rule of exponents states that in any term if the number or expression raised to the power of zero, then the value is always equal to $1$ .

  $a^0=1$

Substitute $1$ for $a^0$ in the expression $\frac{a^0}{a^n}$ .

  $\frac{a^0}{a^n}=\frac{1}{a^n}$

Therefore, the simplified form of expression $\frac{a^0}{a^n}$ by using a zero exponent is $\frac{1}{a^n}$ .

(c).

To determine

To justify: The results in part (a) and (b) for a negative exponent.

Explanation of Solution

Given information: The property to be used is negative exponent.

Calculation:

The simplified form of expression as calculated in part (a) is:

  $\frac{a^0}{a^n}=a^{-n}$

The simplified form of expression as calculated in part (b) is:

  $\frac{a^0}{a^n}=\frac{1}{a^n}$

On comparing the both the simplified form:

  $a^{-n}=\frac{1}{a^n}$

Therefore, it can be explained that the negative sign on an exponent is the reciprocal with positive sign on the exponent.