Property Proofs

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.

Second Property

\frac{a^m}{a^n} = a^{m-n}

For every $a \neq 0$ and $m, n$ real numbers.

Third Property

(a^m)^n = a^{m \cdot n}

For every $a \neq 0$ and $m, n$ real numbers.

Fourth Property

(ab)^m = a^m \cdot b^m

For every $a, b \neq 0$ and $m$ integer.

Proof:

First method:

(ab)^m = \underbrace{ab \times ab \times ab \times \ldots \times ab}_{m\text{ factors}}

Second method:

We can take several examples with specific values of $m, a$ and $b$ to observe the pattern that forms. For example, with $m=2, a=3, b=4$ :

(3 \cdot 4)^2 = 12^2 = 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.

Proof:

\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}}

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 $n \neq 0$ .

Proof:

Using Property $1$ , we obtain:

(a^{\frac{m}{n}})(a^{\frac{p}{n}}) = a^{\frac{m}{n} + \frac{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 $n, q \neq 0$ .

Proof:

Using Property $1$ , we need to equalize the denominators of the exponents:

(a^{\frac{m}{n}})(a^{\frac{p}{q}}) = a^{\frac{m}{n} + \frac{p}{q}}

Example Problems

Determining Values and Proofs

$(3^4)^2 = 3^p$
Solution: Using Property $3$ , $(a^m)^n = a^{m \cdot n}$
$(3^4)^2 = 3^{4 \cdot 2} = 3^8$
$b^4 \cdot b^5 = b^9$
Solution: Using Property $1$ , $a^m \cdot a^n = a^{m+n}$
$b^4 \cdot b^5 = b^{4+5} = b^9$
$(3\pi)^p = 27\pi^3$
Solution: Converting $27\pi^3$ to $3^3 \cdot \pi^3$
$27\pi^3 = 3^3 \cdot \pi^3 = (3\pi)^3$

Simplifying Expressions

$\left(\frac{2^4 \times 3^6}{2^3 \times 3^2}\right)^3$ $\left(\frac{2^4 \times 3^6}{2^3 \times 3^2}\right)^3 = \left(2^{4-3} \times 3^{6-2}\right)^3$
$(3u^3v^5)(9u^4v)$ $(3u^3v^5)(9u^4v) = 3 \cdot 9 \cdot u^{3+4} \cdot v^{5+1}$
$\left(\frac{n^{-1}r^4}{5n^{-6}r^{-4}}\right)^2$ $\left(\frac{n^{-1}r^4}{5n^{-6}r^{-4}}\right)^2 = \left(\frac{n^{-1}r^4}{5} \cdot \frac{1}{n^{-6}r^{-4}}\right)^2$

Exponential Properties and Their Proofs

First Property

a^m \cdot a^n = a^{m+n}

For every $a \neq 0$ and $m, n$ real numbers.

Second Property

\frac{a^m}{a^n} = a^{m-n}

For every $a \neq 0$ and $m, n$ real numbers.

Third Property

(a^m)^n = a^{m \cdot n}

For every $a \neq 0$ and $m, n$ real numbers.

Fourth Property

(ab)^m = a^m \cdot b^m

For every $a, b \neq 0$ and $m$ integer.

Proof:

First method:

(ab)^m = \underbrace{ab \times ab \times ab \times \ldots \times ab}_{m\text{ factors}}

Second method:

We can take several examples with specific values of $m, a$ and $b$ to observe the pattern that forms. For example, with $m=2, a=3, b=4$ :

(3 \cdot 4)^2 = 12^2 = 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.

Proof:

\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}}

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 $n \neq 0$ .

Proof:

Using Property $1$ , we obtain:

(a^{\frac{m}{n}})(a^{\frac{p}{n}}) = a^{\frac{m}{n} + \frac{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 $n, q \neq 0$ .

Proof:

Using Property $1$ , we need to equalize the denominators of the exponents:

(a^{\frac{m}{n}})(a^{\frac{p}{q}}) = a^{\frac{m}{n} + \frac{p}{q}}

Example Problems

Determining Values and Proofs

$(3^4)^2 = 3^p$
Solution: Using Property $3$ , $(a^m)^n = a^{m \cdot n}$
$(3^4)^2 = 3^{4 \cdot 2} = 3^8$
$b^4 \cdot b^5 = b^9$
Solution: Using Property $1$ , $a^m \cdot a^n = a^{m+n}$
$b^4 \cdot b^5 = b^{4+5} = b^9$
$(3\pi)^p = 27\pi^3$
Solution: Converting $27\pi^3$ to $3^3 \cdot \pi^3$
$27\pi^3 = 3^3 \cdot \pi^3 = (3\pi)^3$

Simplifying Expressions

$\left(\frac{2^4 \times 3^6}{2^3 \times 3^2}\right)^3$ $\left(\frac{2^4 \times 3^6}{2^3 \times 3^2}\right)^3 = \left(2^{4-3} \times 3^{6-2}\right)^3$
$(3u^3v^5)(9u^4v)$ $(3u^3v^5)(9u^4v) = 3 \cdot 9 \cdot u^{3+4} \cdot v^{5+1}$
$\left(\frac{n^{-1}r^4}{5n^{-6}r^{-4}}\right)^2$ $\left(\frac{n^{-1}r^4}{5n^{-6}r^{-4}}\right)^2 = \left(\frac{n^{-1}r^4}{5} \cdot \frac{1}{n^{-6}r^{-4}}\right)^2$