Question

Prove the following Laws of Exponents for the case in which m and n are positive integers and $m>n$.

1. Law 2:$\frac{a^m}{a^n}=a^{m-n}$
2. Law 5:${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$

Short answer

1. $\begin{array}{l}\frac{a^m}{a^n}=\frac{\underset{m\text{times}}{\underbrace{a\cdot a\cdot a\cdot a\cdots a}}}{\underset{n\text{times}}{\underbrace{a\cdot a\cdot a\cdot a\cdots a}}}\\ \frac{a^m}{a^n}=\underset{m-n\text{times}}{\underbrace{a\cdot a\cdot a\cdot a\cdots a}}\\ \frac{a^m}{a^n}=a^{m-n}\end{array}$
   
2. $\begin{array}{l}{\left(\frac{a}{b}\right)}^n=\frac{\underset{n\text{times}}{\underbrace{a\cdot a\cdot a\cdot a\cdots a}}}{\underset{m\text{times}}{\underbrace{b\cdot b\cdot b\cdot b\cdots b}}}\\ {\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}\\ {\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}\end{array}$
   

Step-by-step solution

Part a. Step 1. Power of a number.

The power of a number represents how many times to use the number in a multiplication.

Part a.  Step 2. Expand the expression.

Consider the left side of the expression $\frac{a^m}{a^n}$.

This can be written as:

$\frac{a^m}{a^n}=\frac{\underset{m\text{times}}{\underbrace{a\cdot a\cdot a\cdot a\cdots a}}}{\underset{n\text{times}}{\underbrace{a\cdot a\cdot a\cdot a\cdots a}}}$where$m>n$

Part a. Step 3. Simplify..

After simplifying we get:

$\begin{array}{l}\frac{a^m}{a^n}=\underset{m-n\text{times}}{\underbrace{a\cdot a\cdot a\cdot a\cdots a}}\\ \frac{a^m}{a^n}=a^{m-n}\end{array}$

Hence, proved.

Part b. Step 1. Power of a number.

The power of a number represents how many times to use the number in a multiplication.

Part b. Step 2. Expand the expression.

Consider the left side of the expression ${\left(\frac{a}{b}\right)}^n$.

This can be written as:

${\left(\frac{a}{b}\right)}^n=\frac{\underset{n\text{times}}{\underbrace{a\cdot a\cdot a\cdot a\cdots a}}}{\underset{m\text{times}}{\underbrace{b\cdot b\cdot b\cdot b\cdots b}}}$where$m>n$

Part b. Step 3. Simplify.

After simplifying we get:

$\begin{array}{l}{\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}\\ {\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}\end{array}$

Hence, proved.