Download Lesson 3 MA 15200

Lesson 3
MA 15200
Definition of a root: If n is a natural number greater than 1 and a  0 , then
nonnegative number whose nth power is a. ( n a ) n  a
n
a is the
In the radical expression n a , the symbol
is called the radical sign, a is the
radicand, and n is the index. (If the index is 2, a square root, usually the index is not
written. If the index is 3, the expression is called a cube root. Otherwise, it is an nth
root.)
If n is a natural number greater than 1 and a  0 , then n a is the nonnegative number
whose nth power is a. ( n a ) n  a) If a  0 , sometimes n a is defined, sometimes not.
If a  0, then n a is the non-negative number whose nth power is a. ( n a ) n  a
 
n is odd, then n a is the real number such that n a


If a < 0, and 
n is even, the n a is not a real number

n
a
.
If a is a negative, then a would not be real. If a is positive or zero, then a would be
real. To avoid confusion, we will always assume that a variable represents a
positive value.
Ex 1: Evaluate each.
a ) 4 81 
b)
 3 64 
c)
4
 16 
1

8
Multiplication and Division Properties of Radicals:
d)
3

9 4  6 and 9 4  6
Examine:
16
= 2 and
4
16
2
4
If all expressions represent real numbers,
n
a b  ab and
n
n
a
n
b
n
n
a
(b  0)
b
1
Note: These properties are for multiplication and division. Similar statements are
not true for addition or subtraction. ( n a  b  n a  n b , for example)
Numbers that are squares of positive integers; such as 1, 4, 9, 16, ...; are called perfect
squares. Powers such as x 2 , x 4 , x 6 , ... are also called perfect squares.
Numbers that are cubes of positive integers; such as 1, 8, 27, 64,....; are called perfect
cubes. Powers such as x 3 , x 6 , x 9 ,... are also called perfect cubes.
To simplify a radical expression:
1. Factor the radicand into the greatest perfect factor possible and a 'leftover' factor.
2. Use the multiplication property of radicals to 'separate' into two radicals.
3. Simplify the 'perfect factor' radical.
Ex 2: Simplify each.
18 x 3 
a)
b)
3
81a 7b5 
Racial expressions with the same index and the same radicand are called like radicals.
Like radicals can be combined similar to like terms. *When trying to combine radicals,
always simplify radicals first!
Ex 3: Simplify and combine where possible.
a)
32  162 
b)
3a 3a  48a 3 
c)
4 3 16a 4  3a 3 54a 
2
Rationalizing Denominators: Rationalizing a denominator translates ‘write the
expression without a radical in the denominator’. To rationalize when there is a square
root in the denominator, do the following steps.
1. First, simplify each radical.
2. Multiply numerator and denominator by the radical factor in the denominator.
3. Simplify numerator and denominator.
Ex 4: Rationalize the denominator.
2
a)

5
b)
c)
3

8
4

3m3
Definition of a Rational Exponent (1):
1
If a is a real number and n is a natural number, then then a n is the real number b such that
1
b n  a , if b exists. Basically, a n is the same as n a .
Ex 5: Evaluate each.
1
a)
92 
1
3
b)
125 
c)
 81 
 
 16 
d)
 18 
e)
(  4) 2 

1
4

1
1
3
Ex 6: Evaluate each.
1
a)
(4 x 4 ) 2 
1
6 6
b)
(64 x ) 
c)
( 1000 x3 ) 3 
1
Other Rational Exponents (2):
1
m
Definition: If m and n are positive integers, the fraction
is in lowest terms, and a n
n
m
exists, then a n  n a m or (n a ) m .
m
This definition shows that a n can be evaluated two different ways;
(1) take the nth root first, then raise to the mth power or
(2) raise to the mth power first, then take the nth root.
It is usually to the student's advantage to take the root first, then ‘raise to the power’.
Ex 7: Evaluate each.
3
2
a)
36 
b)
83 
4
c)
3
4
16 
4
All the rules of exponents previously studied now apply also with rational exponents.
Ex 8: Assume that all variables represent positive integers. Evaluate or simplify each
and write all answers without using negative exponents.
a)
4

3
2


2
5
b)
(  32)
c)
 8 
 3 6
a b 


2
3

3
92
d)
3
(8 x )

1
3

5
e)
2
(64 x 6 y 12 ) 6 ( x 6 y 3 ) 3 
Ex 9: Simplify each using the rules of exponents. Assume that all variables represent
positive integers. Write answers without using negative exponents.
c c 
1
6 4 2
a)

1
2
c c
3
2


a 

b)
5 

 a2 


3
4

4
3

1
c)
( 16r 4 s 2 ) 2
rs
3
2

5