Download Rules for Operations with Exponents

COLLEGE PREP
SECTION 5A – GETTING READY FOR CHAPTER 5
Objectives:
• Simplify exponential equations using the product rule, the quotient rule, the power rule, and the Law of
Exponents.
• Evaluate Exponential expressions with a Zero or negative exponent.
• Convert between Scientific Notation and Decimal Notation.
• Use Scientific Notation to multiply and divide.
NOTATION:
in the expression , is called the base, and is called the exponent or power.
Rules for Operations with Exponents
Operation
Multiplying – add exponents
Dividing – subtract exponents
Power to a power – multiply exponents
Power of a product – exponent applies to each
factor (like distributing)
Power of a quotient – exponent applies to
numerator and denominator (like distributing)
Power of a negative quotient – exponent applies to
numerator and denominator (like distributing)
This will cause everything inside to switch places.
Negative exponents – moving the exponential factor
to the denominator creates a positive exponent
Zero Exponents – any number or variable that has a
zero exponent is always equal to 1
Formula
Example
· · 2 16 5 125
5
5
5
!
or
"!
$ 1
3 3
$ 1
4
4
Note: These power rules assume that the variable does not equal 0 whenever it’s in the denominator or if
it is raised to the zero power.
BEWARE of these common mistakes!!!
Error Formula
Description
Example
& ' & Exponents do not distribute over addition or subtraction!
3 & 5 ' 3 & 5
& '
& Like terms in fractions do not cancel! (Only factors cancel.)
2&5 2
'
3&5 3
$ ' 0
Zero exponent does not mean the same as multiply by
zero!
' )
Negative exponents do not make a monomial negative!
5$ ' 0
4 ' )4
Simplify each expression
EXAMPLES:
Product Rule:
A)
*+ · **
Remember to deal with the coefficients separately.
3 3
243
B)
+-+ · .-/ 2 · 5 · 0 100 1
Coefficients (numbers) are divided, exponents are subtracted.
Quotient Rule:
C)
2/
2
6 6 216
D)
+.34
5.3*
· 6 6
Zero Exponent Rule: Anything with an exponent of zero should be changed to a 1
E)
.7 1
F)
5487 181 18
Negative Exponent Rule:
Move ONLY the variable that the exponent is attached to. If it’s outside
parentheses, move everything within the parentheses.
G)
+* I)
+
8 *
:
> ":
":
:
>:
>
H)
.;/ J)
. *
*
<=
?
@
· ) +7 -/ · $ · 0 0
Power Rule: If you raise a power to a power, you are multiplying it by itself, therefore, you must raise any
coefficient to the power outside the parentheses, and multiply all exponents.
A+ 7 7·$ 7$ 1
*+ / 3 · 3 · 3 · 3 3· 3 L)
K)
M)
+C/ 2 16
O)
- *
.
F:
:
F:
N)
)/;* + )4 · · P)
.;*
G
+
+ D < :·D
H D·D
D < E
1< E
< E
H=
SCIENTIFIC NOTATION. A number is written in scientific notation when it is in the form
I 10
where 1 J || J 10 and is an integer.
To change a decimal to scientific notation:
Step 1:
Count the number N of decimal places that the decimal point must be moved in order to get
only one digit () in front of the decimal.
Step 2:
If you had to move the decimal to the left (you started with a large number), then your
exponent is positive ( I 10L ). If you had to move the decimal to the right (you started
with a decimal), then your exponent will be negative ( I 10L .
Examples: Write the following in scientific notation.
Q)
+*4, /77 2.384 I 10
R)
7. 7A5 7.1 I 10
Move the decimal 5 places to the left (the original number is
greater than 1), so the exponent is +5.
Move the decimal 2 places to the right (the initial number is
less than 1), so the exponent is -2.
REVERSING THE PROCESS (going from scientific notation to decimal notation):
Look at the exponent on the 10. If the exponent is negative, move the decimal N spaces to the left (toward
the negative end of the number line). If the exponent is positive, move the decimal N spaces to the right (toward the
positive end of the number line).
Examples:
Write the following in decimal notation.
S)
)+. 4 I 57/ )28,000
The positive exponent means the decimal moves to the
right.
T)
5. /? I 57. 0.0000149
The negative exponent moves the decimal to the left.
MULTIPLYING & DIVIDING WITH SCIENTIFIC NOTATION.
Follow the usual rules of exponents, except separate the pieces. Simplify the numbers, then add/subtract the
exponents on the 10’s.
Examples:
U)
P* I 57+ QP+ I 57/ Q 3 · 2 I 10 · 10 6 I 101
V)
P*. + I 57* QP/. 4 I 57/ Q 3.2 · 4.8 I 10 · 10 15.36 I 10 1.536 I 101
W)
+.4I57?
5./I57/
X)
*.2I57*
A.+I57"5
Homework:
.
. I 10@ 2 I 10
.1
. I 10 I 10 .5 I 10 5 I 10
page 351: # 19-24, 31, 33, 37, 41, 45, 51, 57, 65, 69, 71, 81, 83, 85, 87, 91, 101, 105, 115,
117, 121, 131.