Problem (Page 88). In a group of nine coins, eight weigh the same ...

48 3. WHOLE NUMBERS: OPERATIONS AND PROPERTIESExponents – used to simplify multiplication expressions.Definition (Whole Number Exponent).Let a and m be any whole numbers where m 6= 0. Thena m = a · a · · · a | {z }m factorsm is the exponent or power of a, and a is called the base.Example.Theorem.3 5 · 3 3 = (3 · 3 · 3 · 3 · 3) · (3 · 3 · 3) = 3 8Let a, m, and n be any whole numbers where m and n are nonzero. ThenProof.a m · a n = a · a · · · a | {z }m factorsa m · a n = a m+n .· a · a · · · a | {z }n factors= a · a · · · a | {z }m + n factors= a m+n ⇤Example.3 3 · 5 3 = (3 · 3 · 3) = (5 · 5 · 5) = (3 · 5) · (3 · 5) · (3 · 5) = (3 · 5) 3Theorem.Let a, b, and m be any whole numbers where m 6= 0. ThenProof.a m · b m = a · a · · · a | {z }m factorsa m · b m = (ab) m .· b · b · · · b | {z }m factors= (ab) · (ab) · · · (ab)| {z }m factors= (ab) m ⇤