a
n
n
a
Properties and Rules for Exponents
a n a m  a nm
Product
Rule
Bases are
the same
Quotient
Rule
Power
Rule
Properties and Rules for Radicals
an
nm

a
am
Principal
square root of a
a0
a 
1
n
 
m
n
Power of a
Product
Bases are
different
Power of a
quotient
1 1
n
;

a
an an
 ab 
m
m
a
a b
a
a
   m
b
b
a
1
n
b0
a  b if b3  a
 a
n
m
n
a  b if b n  a
nth root of an
if n is an even and positive integer
n
a
| a |
if n is an odd and positive integer
 mn
a
m


a ≥0
Cube root of a
n
m m
m
am 
 a
nth root of a
a 
a 1
0
an 
n
Negative
square root of a
3
n
Negative
Exponent
Rule
a ≥0
Connections
m
m is the power or
a n exponent
 an m
Zero Exponent
Rule
a
n
1
 a
n
Product Rule
for Radicals
m
Quotient Rule
for Radicals
Like radicals
Conjugates
an  a
We can only
multiply/divide
radicals with
the same
root/index.
Properties and Rules for Radicals
Product Rule
for Radicals
n
a
n
index or root
n
a
radicand
Quotient Rule
for Radicals
Like radicals
Conjugates
n
b
n
a

b
n
ab
a
n
b
b0
Radicals with the same radicand
and index/root. We can only
add/subtract like radicals.
Simplifying Radicals
Perfect Squares: 1, 4, 9, 16,
25, 36, 49, 64, 81, 100, 121,
144, 169
Perfect Cubes: 1, 8, 27, 64,
125, 216, 343, 512, 729,
1000
Perfect 4ths: 1, 16, 81, 256,
625
Definition: Numbers
whose roots are whole
numbers. Look for
perfect powers when
trying to simplify roots.
Simplify:
18
Perfect 5ths: 1, 32, 243, 1024
3
40
4
162
3 2
9
2
8
5
81
2
2 5
3
3 2
4
Simplify each radical
81y  9y
5
Hints:
2
y
9
y·y·y·y·y
3
3

2
a
b
3b
24a b
3
9 4
8·3
a9; b3b
When there are variables and
numbers in the problem, simplify
separately
If the root divides evenly into the
power of the variable, it is a perfect
root. You can take out how many
of the variable divide into the root.
Whatever is left over stays under
the radical.
" n " bring out 1”
“For every ______,
4
32z
7
16·2
z4; z3
 2z 2z
4
3
Rational Exponents
16
1
2
1
n
 16  4
a na
1
3
(8)  3 8  2
1
5
(abc) 
1
16 2
(25 x )
5
When a is nonnegative, n can be
any number greater than 1. When
a is negative, n can be any odd
natural number greater than 1
abc
1
2
8
8

25

x

5
x
 25 x
8
Positive Rational Exponents
27
2
3
3
2
 ( 3 27 ) 2  32
(25)  5
3
3
 125
9
4
( 9 )
5
a
4
3
( 7 xy )  (7 xy )
4
For any natural numbers m and n
(n ≠ 0) and any real number a for
which n a exists
5
4
m
n
means ( n a ) m , or
n
am
Negative Rational Exponents
27
1
9
1

9
 9 3
1
2
For any rational number m/n and
any nonzero real number a for
m
which
n
a
a
m
n
means
1
a
(5 xy )
4
5

1
(5 xy )
4
5
m
n
Section 7.3
and 7.4
Multiplying Radicals
Rules to follow:
To multiply radicals, the index must be the same.
Multiply the values inside and outside the radical
separately. If possible, simplify the final answer.
7  2  14
4 2  3 5  12 10
3
3

5 3 6 2
3

 3 30  2 5
Use the distributive property
Dividing Radicals
Rules to follow:
To divide radicals, the root must be the same. Use
the quotient property and write under a single
radical. Simplify the fraction (divide). If possible,
simplify the final answer.
75
 25  5

3
Write under a single
75
3
radical and simplify
3
5 162 x
3
3x
2
8
8
3
6
23
162
x

5
54
x

15
x
2
3
5
2
27•2
3•5
3x
x
Write under a single
radical and simplify
6
You try: Multiply or divide
4
x 6x
2 4
80 y
3x
4
 x 6 x 3x
y
2
2
4
1 80 y

3 5y
3 5y
3
x
3
4
3y

2x
3
x
24
18 x
1
4y y
3

16 y 
3
3
Adding and Subtracting Radicals
Rules to follow: To add/subtract radicals, they must be
like radicals (same root and radicand). Simplify if
possible to make radicands the same. Combine ONLY
the values outside the radical, the radicand does not
change.
5 15 x  2 15 x  7 15x
3
24  4 192  3
3
8•3
3
64•3
Is there a hint
about what
the common
radicand is?
2 3  16 3  3  13 3
3
3
4•4
3
3
2-16+1
You try: Add or Subtract
2 9 y  3 9 y   9y
6 10  4 10
3
50 y  5 y 18
2
 3 y
 6 10  4 10
3
 5 y 2  15 y 2
 20 y 2
You try: Add or Subtract
2 75 9 3


18
18
For this, you must find
10 3 9 3
3
a common radicand



AND a common
denominator.
18
18
18
75
3

9
2
3
8 y  y 27 y
5
3
Find a common
radicand first
by simplifying
the radical
2
 2 y y  3y y
3
2
3
 5y y
3
2
2
Extensions


3 5

x 1  2
3 5

2

 3  5  2

x 1

2
 x 1
 x 1 2 x 1  2 x 1  4
 x  4 x 1  5