48
3. WHOLE NUMBERS: OPERATIONS AND PROPERTIES
Exponents – used to simplify multiplication expressions.
Definition (Whole Number Exponent).
Let a and m be any whole numbers where m 6= 0. Then
am = a
| · a{z· · · a}
m factors
m is the exponent or power of a, and a is called the base.
Example.
Theorem.
35 · 33 = (3 · 3 · 3 · 3 · 3) · (3 · 3 · 3) = 38
Let a, m, and n be any whole numbers where m and n are nonzero. Then
Proof.
am · an = am+n.
m+n
am · an = a
| · a{z· · · a} · a
| · a{z· · · a} = a
| · a{z· · · a} = a
m factors n factors m + n factors
⇤
Example.
33 · 53 = (3 · 3 · 3) = (5 · 5 · 5) = (3 · 5) · (3 · 5) · (3 · 5) = (3 · 5)3
Theorem.
Let a, b, and m be any whole numbers where m 6= 0. Then
Proof.
am · bm = (ab)m.
am · bm = a
· (ab) · · · (ab) = (ab)m
| · a{z· · · a} · b| · b{z· · · }b = (ab)
|
{z
}
m factors m factors
m factors
⇤