Title: Properties of Exponents
1Properties of Exponents
2Definition of Exponent
- An exponent is the power p in an expression ap.
- 52
- The number 5 is the base.
- The number 2 is the exponent.
- The exponent is an instruction that tells us how
many times to use the base in a multiplication.
3Examples
- (4)(4)(4) 64
- (-)(3)(3)(3)(3) -81
- (-2)(-2)(-2)(-2)(-2) -32
- (-3/4)(-3/4) (9/16)
4Puzzler
- (-5)2 -52?
- (-5)(-5) -(5)(5)
- 25 -25
5Multiplication with Exponents by Definition
- 32 35
- (3)(3) (3)(3)(3)(3)(3)
- 37
- Note 257
6Property 1 for Exponents
- If a is any real number and r and s are integers,
then
To multiply two expressions with the same base,
add exponents and use the common base.
7Examples of Property 1
8Examples of Property 1
9Examples of Property 1
10Power to a Power by Definition
- (32)3
- ((3)(3))1((3)(3))1((3)(3))1
- 36
- Note 3(2)6
11Property 2 for Exponents
- If a is any real number and r and s are integers,
then
A power raised to another power is the base
raised to the product of the powers.
12Examples of Property 2
One base, two exponents multiply the exponents.
13Property 3 for Exponents
- If a and b are any real number and r is an
integer, then
Distribute the exponent.
14Examples of Property 3
15Examples of Property 3
16By the Definition of Exponents
Notice that 5 3 2
17Examples
18Negative Exponents
Notice that 3 5 -2
19Definition of Negative Exponents
- If r is a positive integer, then
20Examples of Negative Exponents
Notice that Negative Exponents do not indicate
negative numbers.
Negative exponents do indicate Reciprocals.
21Examples of Negative Exponents
Notice that exponent does not touch the 3.
22Property 4 for Exponents
- If a is any real number and r and s are integers,
then
To divide like bases subtract the exponents.
23Property 5 for Exponents
- If a and b are any two real numbers and r is an
integer, then
The power of a quotient is the quotient of the
powers.
24Property 5 Example
The power of a quotient is the quotient of the
powers.
25Zero as an Exponent
26Zero to the Zero?
Undefined
STOP
Zeros are not allowed in the denominator. So 00
is undefined.
27Examples
28Exponent Rules Summary
Properties
Definitions
29Definition Scientific Notation
- A number is in scientific notation when it is
written as the product of a number between 1 and
10 and an integer power of 10. - A number written in scientific notation has the
form - where 1lt n lt 10 and r an integer.
30Write 1349.7 in scientific notation.
3
31Write 1.85 x 104 in expanded notation.
4
32More on Scientific Notation
How far does the decimal point move?