1. Simplifying Expressions with Integral Exponents
Laws of Exponents
Integral Exponents
Integral here means "integer". So the exponent (or power) is an integer. [That is, either a negative whole number, 0, or a positive whole number.]
Definition: am means "multiply a by itself m times". That is:
am = a × a × a × a × a × ... × a
[We do the multiplication m times.]
Note: This definition only really holds for m > 0, since it doesn't make a lot of sense if m is negative. (You can't multiply something by itself -3 times! And how does multiplying something by itself 0 times give 1?)
In such cases we have to rely on patterns and conventions to define what is going on. See below for zero and negative exponents.
Examples
(1) y5 = y × y × y × y × y
(2) 24 = 2 × 2 × 2 × 2 = 16.
Multiplying Expressions with the Same Base
Definition: am × an = am+n
Let's see how this works with an example.
Example
b5 × b3 |
= (b × b × b × b × b) × (b × b × b) = b8 (that is, b5+3) |
Dividing Expressions with the Same Base
Definition:
(Of course, a ≠ 0)
It may be easiest to see how this one works with an example.
Example
![]()
We cancel 2 of the b's from the numerator and the two b's from the denominator of the fraction. The result is equivalent to b7 − 2.
Repeated Multiplication of a Number Raised to a Power
| (am)n | = (am) × (am) × (am) × ... × (am) [We multiply n times]. |
| = amn |
So we write:
Definition: (am)n = amn
Example
| (p3)2 | = p3 × p3 |
| = (p × p × p) × (p × p × p) | |
| = p6 |
A Product Raised to an Integral Power
Definition: (ab)n = anbn
Example
(5q)3 = 53q3 = 125q3
A Fraction Raised to an Integral Power
Definition: ![]()
Example

Zero Exponents
Definition: a0 = 1 (a ≠ 0)
Example
70 = 1
Note: a0 = 1 is a convention, that is, we agree that raising any number to the power 0 is 1. We cannot multiply a number by itself zero times.
In the case of zero raised to the power 0 (00), mathematicians have been debating this for hundreds of years. It is most commonly regarded as having value 1, but is not so in all places where it occurs. That's why we write a ≠ 0.
Negative Exponents
Definition:
(Once again, a ≠ 0)
In this exponent rule, a cannot equal 0 because you cannot have 0 on the bottom of a fraction.
Example
![]()
Explanations
Observe the following decreasing pattern:
34 = 81
33 = 27
32 = 9
31 = 3
For each step, we are dividing by 3. Now, continuing beyond 31 and dividing by 3 each times gives us:

Summary - Laws of Exponents
[Note: These laws mostly apply if we have fractional exponents, which we meet in the next section, Fractional Exponents.]
Let's try some mixed examples where we have integral exponents.
Example (1)
(a) Simplify ![]()
(b) Simplify ![]()
Example (2)
Simplify ![]()
Example (3)
Here is a LiveMath document to illustrate this.
Now for the normal answer:
Note the following differences carefully:
(-5x)0 = 1, but -5x0 = -5.
Similarly:
(-5)0 = 1, but -50 = -1.
Example (4)
Simplify (2a + b-1)-2
Exercises
Q1 (5an-2)-1
Q2 
Q3 (2a - b-2)-1
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