Download 1.2 – Properties of Exponents

1.2 – Properties of Exponents
• Say you have the an , where a is any real
number and n is any natural number
• Recall, an = a x a x a x a…(n times)
• Repeated multiplication!
• a = base
• n = exponent
Multiplication
•
•
•
•
an x am = an+m
Same base, add
Example; simplify the following
25(210)
• If coefficients are involved, separate; multiply
coefficients, add exponents
• Example; simplify the following
• 5x10(3x5)
Division
m
• a n = am-n
a
• Same base, subtract
• Example
•
x
x
9
4
• If coefficients are involved, separate each part;
divide coefficient, subtract exponents
• Examples; simplify the following
• t
20
t
6
10
• 10 x
5x
6
0 Exponent
• a0 = 1
• This property applies to all expressions AND
numbers raised to the 0 power
• Example; simplify the following
• (x2 + 3x)0 = 1
• 6x0
Negative Exponents
1
•
=
a
• Example; simplify the following
a-m
m
10
2
• When dealing with fractions, simplify “flip”
the term with a negative exponent
• Always look at each term separately
• Example; simplify the following
5
x
z
4
Powers Raised to Powers
• Two scenarios; one with a single term, and
one with multiple terms
n
m
• a   a mn; multiply the powers (no
addition)
• ab  a nbn ; distribute the power to each term,
regardless of the number of terms
n
• NOTE! a  b  a n  bn
n
• Example 1
• Simplify (42)3
• Example 2
• Simplify (3x3)2
• Example 3 [make sure each term is raised to the correct power]
• Simplify (xy2z5)3
Fractions
•
•
n
n
a
a
 
   n
b
b
a
 
b
n
n
bn
 n
a
(reciprocal property)
n
• a n  b n
b
a
•
(“reverse” distribution)
(reciprocal property #2)
If you forget the reciprocal properties, simply use the first in either case
• Example 1
4
• Simplify  2 
3
• Example 2
• Simplify  x 
5
2
• Example 3
• Simplify  xy 
z 
3
4
Multiple Rules
• Often, to simplify an expression, we have to
use multiple rules
• Recall your order of operations!
– PEMDAS
– When dealing with exponents, you often can use
the properties we just covered in multiple ways
• Example
• Simplify (5xy2)-3(3x5z)2
Scientific Notation
• SN is written as a x 10n, where n is an integer
(positive or negative) and 1 ≤ a < 10
• Used in applications of extremely small (mass of
an electron) and extremely large (distance from
the earth to mars) numbers
• Example
• The speed of light in a vacuum is approximately
300,000,000 m/s. Write this number in scientific
notation.
Multiply/Dividing Numbers in Scientific
Notation
• Common Base?
• Follow same rules as mentioned before; add or
subtract exponents; multiply or divide
coefficients
• Example
• (3.6 x 1012)(2.0 x 102)
• What common bases do these numbers share?
• Example
• (8.0 x 1015) ÷ (2.0 x 104)
Assignment
• 1.2
• Pg. 36
• #1-59 odd (skip 49 and 51)