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Learn rigorous proofs of 7 exponential properties with worked demonstrations. Learn multiplication, division, and power rules through examples.
---
## Exponential Properties and Their Proofs
We will discuss the properties of exponents and their proofs. These properties are essential to understand as they form the foundation for solving various problems related to exponents and logarithms.
### First Property
$$a^m \cdot a^n = a^{m+n}$$
For every $$a \neq 0$$ and $$m, n$$ real numbers.
Visible text: For every and real numbers.
### Second Property
$$\frac{a^m}{a^n} = a^{m-n}$$
For every $$a \neq 0$$ and $$m, n$$ real numbers.
Visible text: For every and real numbers.
### Third Property
$$(a^m)^n = a^{m \cdot n}$$
For every $$a \neq 0$$ and $$m, n$$ real numbers.
Visible text: For every and real numbers.
### Fourth Property
$$(ab)^m = a^m \cdot b^m$$
For every $$a, b \neq 0$$ and $$m$$ integer.
Visible text: For every and integer.
**Proof:**
_First method:_
Component: MathContainer
Children:
```math
(ab)^m = \underbrace{ab \times ab \times ab \times \ldots \times ab}_{m\text{ factors}}
```
```math
(ab)^m = \underbrace{a \times a \times a \times \ldots \times a}_{m\text{ factors}} \times \underbrace{b \times b \times b \times \ldots \times b}_{m\text{ factors}}
```
```math
(ab)^m = a^m \times b^m
```
_Second method:_
We can take several examples with specific values of $$m, a$$ and $$b$$
to observe the pattern that forms. For example, with $$m=2, a=3, b=4$$:
Visible text: We can take several examples with specific values of and
to observe the pattern that forms. For example, with :
Component: MathContainer
Children:
```math
(3 \cdot 4)^2 = 12^2 = 144
```
```math
3^2 \cdot 4^2 = 9 \cdot 16 = 144
```
### Fifth Property
$$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$$
For every $$a \neq 0, b \neq 0$$ and $$m$$ integer.
Visible text: For every and integer.
**Proof:**
Component: MathContainer
Children:
```math
\left(\frac{a}{b}\right)^m = \underbrace{\frac{a}{b} \times \frac{a}{b} \times \frac{a}{b} \times \ldots \times \frac{a}{b}}_{m\text{ factors}}
```
```math
\left(\frac{a}{b}\right)^m = \frac{\underbrace{a \times a \times a \times \ldots \times a}_{m\text{ factors}}}{\underbrace{b \times b \times b \times \ldots \times b}_{m\text{ factors}}}
```
```math
\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}
```
### Sixth Property
$$(a^{\frac{m}{n}})(a^{\frac{p}{n}}) = a^{\frac{m+p}{n}}$$
For every $$a > 0$$, $$\frac{m}{n}$$ and $$\frac{p}{n}$$ rational numbers with $$n \neq 0$$.
Visible text: For every , and rational numbers with .
**Proof:**
Using Property $$1$$, we obtain:
Visible text: Using Property , we obtain:
Component: MathContainer
Children:
```math
(a^{\frac{m}{n}})(a^{\frac{p}{n}}) = a^{\frac{m}{n} + \frac{p}{n}}
```
```math
(a^{\frac{m}{n}})(a^{\frac{p}{n}}) = a^{\frac{m+p}{n}}
```
### Seventh Property
$$(a^{\frac{m}{n}})(a^{\frac{p}{q}}) = a^{\frac{mq + pn}{nq}}$$
For every $$a > 0$$, $$\frac{m}{n}$$ and $$\frac{p}{q}$$ rational numbers with $$n, q \neq 0$$.
Visible text: For every , and rational numbers with .
**Proof:**
Using Property $$1$$, we need to equalize the denominators of the exponents:
Visible text: Using Property , we need to equalize the denominators of the exponents:
Component: MathContainer
Children:
```math
(a^{\frac{m}{n}})(a^{\frac{p}{q}}) = a^{\frac{m}{n} + \frac{p}{q}}
```
```math
(a^{\frac{m}{n}})(a^{\frac{p}{q}}) = a^{\frac{mq}{nq} + \frac{pn}{nq}}
```
```math
(a^{\frac{m}{n}})(a^{\frac{p}{q}}) = a^{\frac{mq + pn}{nq}}
```
## Example Problems
### Determining Values and Proofs
1. $$(3^4)^2 = 3^p$$
Solution: Using Property $$3$$, $$(a^m)^n = a^{m \cdot n}$$
```math
(3^4)^2 = 3^{4 \cdot 2} = 3^8
```
```math
p = 8
```
2. $$b^4 \cdot b^5 = b^9$$
Solution: Using Property $$1$$, $$a^m \cdot a^n = a^{m+n}$$
```math
b^4 \cdot b^5 = b^{4+5} = b^9
```
```math
\text{Therefore, the equation is proven true.}
```
3. $$(3\pi)^p = 27\pi^3$$
Solution: Converting $$27\pi^3$$ to $$3^3 \cdot \pi^3$$
```math
27\pi^3 = 3^3 \cdot \pi^3 = (3\pi)^3
```
```math
p = 3
```
Visible text: 1.
Solution: Using Property ,
2.
Solution: Using Property ,
3.
Solution: Converting to
### Simplifying Expressions
1. $$\left(\frac{2^4 \times 3^6}{2^3 \times 3^2}\right)^3$$
```math
\left(\frac{2^4 \times 3^6}{2^3 \times 3^2}\right)^3 = \left(2^{4-3} \times 3^{6-2}\right)^3
```
```math
= \left(2^1 \times 3^4\right)^3
```
```math
= 2^{1 \cdot 3} \times 3^{4 \cdot 3}
```
```math
= 2^3 \times 3^{12}
```
2. $$(3u^3v^5)(9u^4v)$$
```math
(3u^3v^5)(9u^4v) = 3 \cdot 9 \cdot u^{3+4} \cdot v^{5+1}
```
```math
= 3^1 \cdot 3^2 \cdot u^7 \cdot v^6
```
```math
= 3^{1+2} \cdot u^7 \cdot v^6
```
```math
= 3^3 \cdot u^7 \cdot v^6
```
```math
= 27u^7v^6
```
3. $$\left(\frac{n^{-1}r^4}{5n^{-6}r^{-4}}\right)^2$$
```math
\left(\frac{n^{-1}r^4}{5n^{-6}r^{-4}}\right)^2 = \left(\frac{n^{-1}r^4}{5} \cdot \frac{1}{n^{-6}r^{-4}}\right)^2
```
```math
= \left(\frac{n^{-1}r^4}{5} \cdot n^6r^4\right)^2
```
```math
= \left(\frac{n^{-1+6}r^{4+4}}{5}\right)^2
```
```math
= \left(\frac{n^5r^8}{5}\right)^2
```
```math
= \frac{n^{10}r^{16}}{25}
```
Visible text: 1.
2.
3.