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Algebra 2
5.1 Properties of Exponents
5x
Name__________________Date ______
The coefficient is ________. It is ______________________
The base is the _______. It is the ______________________
The exponent is the _______. It is the ___________________
8
When we add a variable over and over, we have
an easier way to write the expression:
x + x = 2x
x + x + x = 3x
x + x + x + x = 4x
Etc.
You try:
When we multiply the same number or variable
over and over, we also have a shortcut!!
2 • 2 = 22
2 • 2 • 2 = 23
2 • 2 • 2• 2 = 24
x
x
x
x
•
•
•
•
x
x
x
x
= x2
• x = x3
• x • x = x4
• y • y • y = x2y3
x • x • x • x • x = ______
a•a•a•a=
______
c • c • c • d • d = ______
Bases must be the same to multiply!!
Now, watch what happens when we multiply two expressions together!
Problem
→→
x2
•
x3
Think
→→
x•x • x• x •x
Answer →→
Think
→→
Problem
Think
x is used as a factor 5 times!!
2 plus 3 = 5
→→
→→
Answer →→
Think
→→
1.
x4
•
x•x•x•x •
x2
x• x
x is used as a factor 6 times!!
4 plus 2 = 6
x2 • x8 =
2. x • x5 =
x = x1)
x5
x6
(don’t forget that
3.
x3 • x6 • x2 =
Instead of writing out the problem the long way, there is a simple rule…
To multiply powers that have the same base, just add the exponents!!!
am • an = am+n
4.
x2 • x7
5.
x3 • x56
6.
x4 • x9 • x3
What if we have coefficients???
Example 1
5x2 y2 • 3x5y
Arrange the factors so that the coefficients are together in the front of the
expression.
Step 1: Multiply the numbers first.
Step 2: Multiply the different variables separately.
5 • 3 • x2 •x5 • y2 • y
Now, simplify
15x7y3
You try!!
7. x • 3x5 • 4x3
8. (xy)(x2y5)
9. 3x2(−2y2)(4xy)
Example.
(x2)3 = x2 • x2 • x2
Something cubed is something times itself 3 times.
= x•x • x•x • x•x
Expand the x-squared’s.
= x6
How many times is x multiplied by itself?
Write out the “long way” and simplify.
10. (x4)2
11.
(c4)3
Rule!
When you have a power of a power, you multiply the outside exponent with all inside exponents!
Try.
12. (3d7)3
13. (2y2)5

y

2y3
A little division!! Write these the “long way.”
15.
x5
x2
16.
14. (-n3)6
x8
x6

n2
Quotient of Powers Property: to divide powers having the same base, reduce the plain numbers
and subtract exponents that have the SAME base.
You try:
17.
3x 5
9x 2
18.
****
x 0 = 1. Memorize this!!!!
****
x n
Try:
20.
means the reciprocal of
6x 0
Watch this:
6x 5 y 3
4x 2 y
21.
x4
x 5
x 7
19.
x2 y
x2
xn . It means ________________________________
22.
x 2 y5
23.
There are two ways to simplify this problem:
a.
Put everything where it belongs first:
b
Use the rule:
Try a few.
24.
26.
10 x9
5 x3
3x 5
27x 4
25.
6x 3

10x
27.
6 x 5
27 x 4
2x 6
6x 2y  4
6x 3y 4

10x
5x 2
QUOTABLE PUZZLES—Exponents
Name _________________________
Directions: Solve the following problems. Match that answer to the correct letter of the
alphabet. Enter that letter of the alphabet on the blank corresponding to the problem
number.
___
7
___
6
___
18
___
19
___
20
___
9
___
12
___
16
___
19
___
11
___
18
___ ___ ___ ___ ___ ___
5
18
A
x4
2
18
B
125x12
J
K
3
9a
x8
R
t6
5
C
1
L
16
S
7x3
___
1
___
12
___
16
___
3
--
E
-125x3
F
4x5
8
20
N
T
-24x5
___
14
___
D
10x5
7x2
___
17
-- ___ ___
M
2a3x2
___
1
10
O
+ 35x
P
5b2
z4
U
-8x9
V
4x6
W
102
___
19
___
14
___
19
___
4
___
15
G
0
___ ,
8
___
5
___
4
___
13
H
x6y9
___
5
___
19
I
x3 y
Q
x6
X
27a3
Y
8x4
10x4
Z
x5y6
Simplify:
1. x2 • x4
11. (3a)3
9. (x2 y3)
2. (2x3)2
12. (a2 x) (2ax)
10. (5x4)3
3. 4x3 • 2x
13.
4.
14.
x6
x2
a0
z10
z6
19. (-8x3) (3x2)
5. 2x • 5x4
15. t2 • t2 • t2
6. 7x(x + 5)
16. (-5x)3
7.
10 6
104
17.
20b3
4b
8.
14x
2x2
18.
x y
x2 y5
5
3
5
6
20. (-2x3)3
Oops on the numbering!!