There are several useful rules for operating on numbers with exponents. The following four rules, also known as "identities," hold for all integer exponents, provided that the base is non-zero.

Multiplying Exponential Expressions with the Same Base  

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

$a^m$ means that you have $m$ factors of $a$. If you multiply this quantity by $a^n$, i.e. by $n$ additional factors of $a$, then you have $a^{m+n}$ factors in total. For example:

$5^3 \cdot 5^4=5^{3+4}=5^7$

Note that you can only add exponents in this way if the corresponding terms have the same base.

Dividing Exponential Expressions with the Same Base

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

In the same way that ${ a }^{ m }\cdot { a }^{ n }={ a }^{ m+n }$ because you are adding on factors of $a$, dividing removes factors of $a$. If you have $n$ factors of $a$ in the denominator, then you can cross out $n$ factors from the numerator. If there were $m$ factors in the numerator, now you have $(m-n)$ factors in the numerator.

In order to visualize this process, consider the fraction:

$\dfrac{3^5}{3^2}$

This fraction can be rewritten as:

$\dfrac{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3}$

Here you can see that two 3s will cancel out from the numerator and denominator. We are left with:

$\dfrac{3 \cdot 3 \cdot 3}{1} = 3^3$

As an additional example:

$\displaystyle \frac{7^4}{7^2}=7^{4-2}=7^2$

Raising an Exponential Expression to an Exponent

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

If you think about an exponent as telling you that you have a certain number of factors of the base, then ${({a}^{n})}^{m}$ means that you have factors $m$ of $a^n$. Therefore, you have $m$ groups of $a^n$, and each one of those has $n$ groups of $a$. Therefore, you have $m$ groups of $n$ groups of $a$; therefore, you have $n \cdot m$ groups of $a$, or ${a}^{n \cdot m}$. For example:

$(3^3)^3=3^{3 \cdot 3}=3^9$

Raising a Product to an Exponent

${(ab)}^{n}={a}^{n}{b}^{n}$

You can multiply numbers in any order you please. Instead of multiplying together $n$ factors equal to $ab$, you could multiply all of the $a$s together and all of the $b$s together and then finish by multiplying $a^n$ by $b^n$. For example:

$(4\cdot5)^3=4^3\cdot5^3$

Example

Simplify the following expression: $(3\cdot2)^3\cdot (2^5)^4$

For the first part of the expression, apply the rule for a product raised to an exponent:

$(3\cdot2)^3\cdot (2^5)^4 = 3^3 \cdot 2^3 \cdot (2^5)^4$

For the last part of the expression, apply the rule for raising an exponential expression to an exponent:

$3^3 \cdot 2^3 \cdot (2^5)^4 = 3^3 \cdot 2^3 \cdot 2^{5\cdot 4} = 3^3 \cdot 2^3 \cdot 2^{20}$

Notice that two of the terms in this expression have the same base: 2. These two terms can be combined by applying the rule for multiplying exponential expressions with the same base:

$3^3 \cdot 2^3 \cdot 2^{20} = 3^3 \cdot 2^{3+20} = 3^3 \cdot 2^{23}$

Therefore, $3^3 \cdot 2^{23}$ is the simplified form of this expression.