Exponential Unit Math 11
The expression 2 3 is called a power. The base is 2 , and the exponent is 3.
Part A: How can you multiply powers with the same base?
Product of
Powers
Expanded Form
Result as a Power
(23 )( 24)
2x2x2x2x2x2x2
27
(103 )(105)
(-2)2(-2)
(x3 )(x3)
(-y)2 (-y)3
Part B: How can you divide powers with the same base?
Quotient of Powers
75 ÷ 72
103 ÷ 103
(-2)4 ÷ (-2)2
x3 ÷ x 2
(-y)7 ÷ (-7)3
Expanded Form
7x7x7x7x7
7x7
Result as a Power
73
Part C: How can you find the power of a power?
Power of a Power
(23)2
Expanded Form I
Expanded Form II
(23 )(23)
2x2x2x2x2x2
Result as a Power
26
(103)4
[(-4)2]2
(x2)3
[(-y)4]2
Part D: How can you find the power of a product?
Power of a Product
(3 x 2)2
Expanded Form I
(3 x 2)(3 x 2)
Expanded Form II
3x3x2x2
Result as a Power
32 x 22
(2 x 5)3
(-4 x 3)2
(x y)5
(-p q)3
Part E: How can you find the power of a quotient?
Power of a Quotient
3
2
 
3
3
 
4
5
x
 
y
4
Expanded Form I
2 2 2
x
x
3 3 3
Expanded Form II
2x2x2
3x3x3
Result of a Power
23
33
The Laws of Exponents
Law 1
x m  x n  x m n
eg. x5  x 2  x5 2  x7
Law 2
x m  x n  x mn
eg. x5  x 2  x52  x3
x   x
eg.  x   x
Law 3
m n
mn
5 3
 xy 
Law 4
eg.
Law 5
n
35
 x15
 xn y n
 xy 
3
 x3 y 3
n
x
xn

 
yn
 y
y0
5
x
x5
eg.    5
y
 y
1. Simplify
a. x5  x 23
b. x12  x 3  x 4
c. m5  m8  m3
d. 9 x5  7 x5
e. 75m18  15m6
f.
g. 12 y 9  8 y12
h. 3c 4
j. m11n8  m5 n 4
k. 7c12 d 5  6c8 d 6
15 12
5 4
m. 36a b  9a b
n. x 2 y 3
 24c8 d 5  15c3d 9 
p. 

2
5 
 8c d  18cd 



i. a 7b 4  a 9b3
5
  x y
a b 
q.
 ab 
2
2 3 4
2 3
24a 20  6a10
2
4
 
l. xy 2
3
 12m3n2  18m5 n3 

2 2 
 9mn  4m n 
o. 
2. Evaluate for x  2 and y  1
2
 4 x y   3x y 
6x y 
3
 3x y  5x y 
a.
4
b.
6
2 2
5
7
2 2
3. Simplify
 2a  3a 
 6a 
3x
a.
2 x 1 4
b.
x 2
x
x
2m
y n 1 
3 m 1
yn 
3
2
4. If x  3m2 and y  m3 , write each expression in terms of m.
b. 5x 3 y 4  3  xy 
a. 2 x 2 y
Integral exponents
Zero Exponent:
Expanding
x3
we get
x3
x x x 
x x x 
which equals 1.
Using the law of exponents, we get x3-3 = x0
Therefore, x0=1 (for all x not equal to zero) .
Negative exponents:
Now, a
n
Therefore,
a
0 n
a0
1
 n  n
a
a
a-n =
1
an
x 0  1 where  x  0  and x  n 
1
xn
 n  R, x  0 
2
9 3
5. Evaluate without a calculator.
 3

a.  2 
1
3
b.  
4
2
 1
c.   
 2
3
3
d.  
4
0
6. Simplify
31  32
a.
b.
 31  32 
4
70
 23 
2
c. 24c d  4c d
5
1
3
8
3
12m5 n 2  5m11n 6
d.
15m3n 4
Rational Exponents
1
n
x  n x where n  N x  0, if n is even
x

1
n

1
where n  N x  0, if n is even
x
n
m
n
x  n xm =
7.
a.
 12 
3
5
m
x  0, if n is even
b.

5

1
17
94
c.

3
25

4
d.

1
52

7
Simplify
2
3
2
a. 25  8 3
 9
4
n
Express as a power
8.
d.
 x
2
2
2
b. 32 3  2435
1
e. 9   
8
1
2

2
3
2
1
c. 64 3 1253
1
4
1
3
f.  125 3  32 5  36 2






2
9. Express as a power with base 2
a. 8
b. 1
c.
d. 16 n
2
f. 83n  42 n
c.
 2  4 
n 3
g.
n
1
16
e.
 2  4 8 
h.
 4  2  16
n 1
n
3 n 1
n2
n 5
8n 1
10. Express as a power with base 3
a.
4
b.
3
 
d. 9 4
1
d.
27
32 x
729
c. 812 x
  9 
e. 3 x 1
2
x
n 1