HISTORY of the word HISTORY of the word EXPONENT.EXPONENT.
The term EXPONENT The term EXPONENT was introduced by was introduced by Michael Stifel (1487-Michael Stifel (1487-1567) in 1544 in 1567) in 1544 in Arithmetica integra. Arithmetica integra.
So 4³ is the same as (4)(4)(4), three identical factors of 4. And x³ is just three factors of x, (x)(x)(x).
ExponentsExponents
35power
base
exponent
3 3 means that is the exponential
form of t
Example:
he number
125 5 5
.125
Using the CalculatorUsing the Calculator
5 5 44
Press 5Press 5
Press Press ^̂Press 4Press 4
Then =Then =
7 Laws of Exponents #17 Laws of Exponents #1PRODUCT LAWPRODUCT LAW
To To MultiplyMultiply LIKE Bases… LIKE Bases…
……Copy the Base, Add Exponents Copy the Base, Add Exponents
Product Law or Product RuleProduct Law or Product Rule
m n m nx x x 3 4 3 4 7Example: 2 2 2 2
3 4
7
Proof: 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
7 Laws of Exponents #27 Laws of Exponents #2QUOTIENT LAWQUOTIENT LAW
To Divide LIKE Bases…To Divide LIKE Bases…
……Subtract ExponentsSubtract Exponents
6
2
a
a a4
Quotient Law or Quotient Rule:Quotient Law or Quotient Rule:
mm n m n
n
xx x x
x
44 3 4 3 1
3
5Example: 5 5 5 5 5
5
4
3
5 5 5 5 5Proof: 5
5 5 5 5
7 Laws of Exponents #37 Laws of Exponents #3EXPONENT of EXPONENT LAWEXPONENT of EXPONENT LAW
To Raise a Power to a Power…To Raise a Power to a Power…
……Multiply ExponentsMultiply Exponents
43a a12
Exponent of Exponent Law or Exponential Rule:Exponent of Exponent Law or Exponential Rule:
nm mnx x
23 3 2 6Example: 4 4 4
2 23
6
Proof: 4 4 4 4 4 4 4 4 4 4
4 4 4 4 4 4 4
To Raise a QUANTITY To Raise a QUANTITY to a Power, raise EACH to a Power, raise EACH Factor to that Power.Factor to that Power.
(-3ab)2= 9a2b2
7 Laws of Exponents #47 Laws of Exponents #4Raising a product to a Raising a product to a
powerpower
7 Laws of Exponents #57 Laws of Exponents #5Raising a quotient or a Raising a quotient or a
fraction to a powerfraction to a power To Raise a FRACTION To Raise a FRACTION
to a Power, raise BOTH to a Power, raise BOTH Numerator & Denominator Numerator & Denominator to that power. to that power.
4a
b
a4
b4
7 Laws of Exponents #67 Laws of Exponents #6NEGATIVE EXPONENT LAWNEGATIVE EXPONENT LAW
Negative Exponents Negative Exponents
……Reciprocal with a Positive ExponentReciprocal with a Positive Exponent
3a a3
#6: Negative Law of Exponents: If the base is powered by the negative exponent, then the base becomes reciprocal with the positive exponent.
1mm
xx
33
1 1Example #1: 2
2 8
33
3
1 5Example #2: 5 125
5 1
7 Laws of Exponents #77 Laws of Exponents #7
Any nonzero number raised to the ZERO Power = Any nonzero number raised to the ZERO Power = ONEONE
0a
02
03,263,546
The Laws of Exponents:The Laws of Exponents:
#7: Zero Law of Exponents: Any base powered by zero exponent equals one
0 1x
0
0
0
Example: 112 1
51
7
1flower