Chapter 4.6, Problem 61E

i.

To determine

To identify: The quotient $\frac{a^n}{a^n}$ using the powers property.

Answer to Problem 61E

  $\frac{a^n}{a^n}=1$

Explanation of Solution

Given information:

The quotient,

  $\frac{a^n}{a^n}$

Use power formula,

  $\begin{array}{l}\frac{a^n}{b^m}=a^n{b}^{-m}\\ \text{and}\\ \frac{a^n}{a^m}=a^{n-m}\end{array}$

Then,

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

Since $a^0=1$

So,

  $\frac{a^n}{a^n}=1$

Conclusion:

To simplify the quotient $\frac{a^n}{a^n}$ and get the result is equal to 1.

ii.

To determine

To identify: The solution of quotient $\frac{a^n}{a^n}$ in different way.

Answer to Problem 61E

Different ways to simplify the given quotient is

  $\frac{a^n}{a^n}=\frac{\frac{a^{n+1}}{a^n}}{\frac{a^{n+1}}{a^n}}$

And get the formula of exponent is

  $a^n=\frac{a^{n+1}}{a^n}$

Explanation of Solution

Given information:

Quotient $\frac{a^n}{a^n}$

Concept Involved:

Use power formula,

  $\begin{array}{l}\frac{a^n}{b^m}=a^n{b}^{-m}\\ \text{and}\\ \frac{a^n}{a^m}=a^{n-m}\end{array}$

Multiply the quotient $\frac{a^n}{a^n}$ by a , get the result

  $\begin{array}{c}\frac{a^n}{a^n}=\frac{a^n\times a^1}{a^n\times a^1}\\ =\frac{a^{n+1}}{a^{n+1}}\end{array}$

Divide by the common factor of $a^n\text{and}{a}^{n+1}$ is $a^n$ in the numerator and denominator.

  $\frac{a^{n+1}}{a^{n+1}}=\frac{\frac{a^{n+1}}{a^n}}{\frac{a^{n+1}}{a^n}}$

Therefore,

  $\frac{a^n}{a^n}=\frac{\frac{a^{n+1}}{a^n}}{\frac{a^{n+1}}{a^n}}$

Get the definition of exponent is

  $a^n=\frac{a^{n+1}}{a^n}$

Conclusion:

Different ways to simplify the given quotient is

  $\frac{a^n}{a^n}=\frac{\frac{a^{n+1}}{a^n}}{\frac{a^{n+1}}{a^n}}$

And get the formula of exponent is

  $a^n=\frac{a^{n+1}}{a^n}$

iii.

To determine

To identify: The solution of quotient $\frac{a^0}{a^n}$ .

Answer to Problem 61E

  $\frac{a^0}{a^n}=a^{0-n}$

Explanation of Solution

Given information:

Quotient $\frac{a^0}{a^n}$

Concept Involved:

Use power formula,

  $\begin{array}{l}\frac{a^n}{b^m}=a^n{b}^{-m}\\ \text{and}\\ \frac{a^n}{a^m}=a^{n-m}\end{array}$

  $\frac{a^0}{a^n}=a^0\times a^{-n}$

Then,

  $\frac{a^0}{a^n}=a^{0-n}$

Conclusion:

To simplify the expression $\frac{a^0}{a^n}$ and get the result $a^{0-n}$ .

iv.

To determine

To identify: The solution of quotient $\frac{a^0}{a^n}$ using the definition of zero exponent.

Answer to Problem 61E

  $\frac{a^0}{a^n}=a^{-n}$

Explanation of Solution

Given information:

Quotient $\frac{a^0}{a^n}$

Concept Involved:

Use power formula,

  $\begin{array}{l}\frac{a^n}{b^m}=a^n{b}^{-m}\\ \text{and}\\ \frac{a^n}{a^m}=a^{n-m}\end{array}$

  $\frac{a^0}{a^n}=a^0\times a^{-n}$

Since, by the definition of zero exponent

  $a^0=1$

Then,

  $\begin{array}{c}\frac{a^0}{a^n}=1\times a^{-n}\\ =a^{-n}\end{array}$

So, get the definition of exponent is

  $\frac{a^0}{a^n}=a^{-n}$

Conclusion:

To simplify the expression $\frac{a^0}{a^n}$ and get the result $a^{-n}$ .