A number raised to a power represents a product where the same number is used as a repeated factor. The number is called the base and the power is given by the exponent. The base is the repeated factor (the number multiplied) and the exponent counts the number of factors. An exponent means that we are dealing with products and multiplication.

In the expression bⁿ, b is the base and n is the exponent.

This expression means we use b as the factor and we have n factors of b. For example:

5³ (read five to the third power) means we have 3 factors of 5, or 5*5*5 which simplifies to 125.

5³ is the exponent form,
5*5*5 is the expanded form, and
125 is the product or simplified form.

Exponent Form	Expanded Form	Simplified (product Form)
53	5*5*5	125
35	3*3*3*3*3	243
92	9*9	81
34	3*3*3*3	81
x3	x*x*x	x3

 

When we calculate using numbers in exponent form that have the same base, we can always convert to expanded form, count the number of factors, then change back to exponent form, especially when the base is a variable. But this is a pain, so mathematicians have developed shortcuts, called RULES, to make the calculations quicker and easier to write.

Examples

Multiply x³ times x⁵:
We could expand to (x*x*x) * (x*x*x*x*x), then count the factors of x and convert back to exponent form. Since there are now 8 factors of x, we write x⁸.
Where did the 8 come from? Well, we have 3 factors of x for the x³ and 5 factors of x for x⁵, and that adds to 8 factors of x. Since x is still our base and our new exponent is 8; we can write our product as x⁸.
When we multiply two numbers having the same base, we can add the original exponents to find the new exponent of the product. This sounds like a shortcut (AKA: RULE):

The Product Rule for Exponents: a^m * aⁿ = a^{m + n}.

Divide x⁷ by x⁴:
Expand to ^{x*x*x*x*x*x*x}/_^x*x*x*x. One  x in the numerator (on the top) will divide to 1 with one of the x’s in teh denominator (on the bottom) until there are no more x’s on the bottom, leaving 3 x’s on top over a 1 on the bottom: , which simplifies to or x³.
We also notice that 7 – 4 = 3, which is our shortcut (Rule) to find our quotient.

The Quotient Rule for Exponents: a^m / aⁿ = a^m–n.

Find (x³)⁴:
Expand to (x³)*(x³)*(x³)*(x³). Now apply the product rule: x^3+3+3+3 = x¹².
Notice also that 3*4 = 12. We can multiply the exponent by the power to simplify, so we have a shortcut (Rule) to find our power:

The Power Rule for Exponents: (a^m)ⁿ = a^m*n.

Find x^-2:
Remember the Quotient Rule: x^m / xⁿ = x^m-n.
What happens when n is > m? You get a negative exponent. Let’s see what this looks like in expanded form:

If we apply the Quotient Rule, we get x^3–5 = x^–2.
Therefore, x^–2 = 1/x²

Negative Exponent Rule: x^–n = 1/xⁿ.

How do we evaluate x⁰?

Again, it goes back to the Quotient Rule: Find x³/x³.

Zero Exponent Rule: x⁰ = 1, for all x ≠ 0.

Summary of Rules (think: shortcuts)
The Product Rule for Exponents: a^m * aⁿ = a^{m + n}.
To find the product of two numbers with the same base, add the exponents.

The Quotient Rule for Exponents: a^m / aⁿ = a^m–n.
To find the quotient of two numbers with the same base, subtract the exponent of the denominator from the exponent of the numerator.

The Power Rule for Exponents: (a^m)ⁿ = a^m*n.
To raise a number with an exponent to a power, multiply the exponent times the power.

Negative Exponent Rule: x^–n = 1/xⁿ.
Invert the base to change a negative exponent into a positive.

Zero Exponent Rule: x⁰ = 1, for .
Any non-zero number raised to the zeroth power is 1.