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RADICALS
n m
a
1
RADICALS
Upon completion, you should be able to
• define the principal root of numbers
• simplify radicals
• perform addition, subtraction,
multiplication, and division of radicals
Mathematics Division, IMSP, UPLB
2
nth Root
Definition:
Suppose b  a.
n
Then we call
b an nth root of a.
Examples:
 2
3
 8
-2 is the cube root of -8
2   2  4
2
2
2 and -2 are the square
roots of 4
Mathematics Division, IMSP, UPLB
3
NOTICE THAT WE ARE NOT YET
USING HERE THE RADICAL SIGN
nth Root
Definition:
n
Suppose b  a.
n
Then we call
b an nth root of a.
Examples:
 2
3
 8
-2 is the cube root of -8
2   2  4
2
2
2 and -2 are the square
roots of 4
Mathematics Division, IMSP, UPLB
4
nth Root
Notice that if n is odd, then we only have one
nth root.
But if n is even, then we have two nth roots (the
positive nth root and the negative nth root).
Example:
 3  27
2
2
2  4 and  2  4
3  27
3
3
Mathematics Division, IMSP, UPLB
5
Principal nth Root
Definition:
Let n be positive even integer and a > 0,
To remove the ambiguity, we define
n
to be the positive root.
a
We call this the principal nth root.
Hence,
Also,
42
4 256
and not –2.
4
 ____
a is called the radicand
n is the index of the radical.
Mathematics Division, IMSP, UPLB
6
Principal nth Root
FYI: Let n be positive even or odd integer
n
0  0.
Mathematics Division, IMSP, UPLB
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Principal nth Root
In summary, the radical sign
n
denotes the principal nth root of a number.
Mathematics Division, IMSP, UPLB
8
RADICALS
A radical or (irrational expression) is an algebraic expression
involving non-integral rational exponents. It is of the form
n
a
m
or
m
an
Examples: Write as a radical.
1.
2.
1/2 3/4
2a
b
1/3
y
1/2
y
x
x
1/3
1/2
4 3
2 a b

3
x3 y
x y
Mathematics Division, IMSP, UPLB
9
RADICALS
Examples: Write the following radicals in exponential form.
1.
2.
x
1/3 2/3 1/3
2
3
xy z
3
x
4
x
6
y
3
y
5
y
1/3
x
z
y
3/4
 1/2
5/6
x y
Mathematics Division, IMSP, UPLB
10
RADICALS
DEFINITION: If m and n are NOT both even.
a 
1
m n

a


1
n
 a a
m
n

 


m
 a
n
m
m
n
a
Mathematics Division, IMSP, UPLB
m
n
They
are
equal.
11
RADICALS
Examples: If m and n are NOT both even.
odd
even
3
2
 
2  2
1
3 2

2   2

3
2
1
2
 2
3

 


Mathematics Division, IMSP, UPLB
3
 2
3
12
RADICALS
DEFINITION: If m and n are both even.
a 
1
m n

a


1
n
 a a
m
n
m
 
m
n

m
  n a a

 Notice the difference if m is
m
n
inside or outside the parenthesis.
Mathematics Division, IMSP, UPLB
13
RADICALS
Examples: If m and n are both even.
even
 2 
1
2 2
even

 2
2
 4  4  2   2   2
1
2
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2
2
RADICALS
Examples: If m and n are both even.
a a
2
 a
2
4
a
2
4
1
2
x  x  x 
2
Mathematics Division, IMSP, UPLB
x
15
RADICALS
DEFINITION: Negative Rational Exponent
a
m n
1
 m
n
a
Mathematics Division, IMSP, UPLB
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Properties of Radicals
Theorem: If a and b are real numbers, then
1.
n
ab  (ab)
1/ n
1/ n
2.
n
a a
 
b b
a
 n an b
1/ n 1/ n

b
1/ n
a
1/ n
b
n
a
 n , b0
b
(where both a > 0 and b > 0 if n is even).
Mathematics Division, IMSP, UPLB
17
Properties of Radicals
Notice the clause: “where both a > 0 and b > 0
if n is even”. Because:
 1 1  1
1 1 
FYI: sqrt(-1) is an imaginary number
1 1 


2
 1  1
Mathematics Division, IMSP, UPLB
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Properties of Radicals
Examples: Use the properties to find the ff.
1.
24 6
2.
3 5 3 25
3.
4.
4 324
44
33
3
24
Mathematics Division, IMSP, UPLB
19
Properties of Radicals
Theorem:
mn
a
mn
 a
Mathematics Division, IMSP, UPLB
20
Simplifying Radicals
A radical is simplified if the following hold:
1. There is no power in the radicand higher than or equal to
the index.
Examples: Simplify.
1.
2.
48x
3
 2 3 x
4
3
 4 | x | 3x
4 16x16 y 7 z 9  4 24 x 4 4 y 4 y3 z8 z 
 
Mathematics Division, IMSP, UPLB
2 x
4
yz

2 4
21
3
yz
Simplifying Radicals
A radical is simplified if the following hold:
2. The index and the exponents in the radicand must have no
common factor.
Examples: Simplify.
3.
4.
44
 2  2
4 2
r s t  s
6 2 6 4
6
r t  sr t  s
2 4
Mathematics Division, IMSP, UPLB
2
6
4
6
3
rt
22
2
Simplifying Radicals
A radical is simplified if the following hold:
3. There is no denominator in the radicand.
Examples: Simplify.
5.
2x
3y
6 3
6.
8x y
15 z 7
2x

3y
3
23 x 6 y 3  2 x y 2 y

3
7
z 15 z
15 z
Mathematics Division, IMSP, UPLB
23
Multiplying Radicals
RECALL:
n
n
a b
 ab
n
Examples: Find these products and simplify.
1.
4
2.
3 4 x 2 y 2 3 6 x 2 z 2 3 45 x 2 y 2 z
24 x
34
270 x
2
Mathematics Division, IMSP, UPLB
Assume
variables are
greater than
zero.
24
Multiplying Radicals
How do you multiply radicals with different indices?
Examples: Find these products and simplify.
3
1.
2 2
2.
3
2 4
3.
x3
y
24
Assume
variables are
greater than
zero.
z
Mathematics Division, IMSP, UPLB
25
Dividing Radicals
RECALL:
n
n
a
a
 n , b0
b
b
Examples: Find the quotients and simplify.
1.
2.
4 x3 y 2
Assume
variables are
greater than
zero.
34 xy
8 x6 y3
15 z 7
Mathematics Division, IMSP, UPLB
26
Dividing Radicals
What if the indices are not the same?
Examples: Find the quotients and simplify.
Assume
variables are
greater than
zero.
2
1. 3
3
4 3
x
2.
3 xy
Mathematics Division, IMSP, UPLB
27
Rationalizing the Denominator
To rationalize the denominator means to get rid of radicals in
the denominator.
Multiply numerator and denominator by a rationalizing factor.
Examples: Rationalize the denominator.
3
1.
2 3
2 2 3 7
2.
3 22 7
Mathematics Division, IMSP, UPLB
28
Rationalizing the Denominator
Examples: Rationalize the denominator.
2
3.
4.
25 x yz
2
3 100 x 4 y 5 z 2
Assume
variables are
greater than
zero.
a b
a b
Mathematics Division, IMSP, UPLB
29
Rationalizing the Denominator
Examples: Rationalize the denominator.
1
5. 3
x3 y
6.
Assume
variables are
greater than
zero.
1
3 2
b
3
 bc
3 2
 c
Mathematics Division, IMSP, UPLB
30
Similar Radicals
Radicals are similar if they have the same index and radicands
when simplified.
Examples: Which of the following pairs of radicals are similar?
Assume x>0
1.
3
48 x , 12 x
3.
4
2.
3 2x , 6
4x
2
1
5
, 8x
2x
Mathematics Division, IMSP, UPLB
31
Adding and Subtracting Radicals
We can only add(subtract) similar radicals. To do that,
add(subtract) their coefficients and affix the common radical.
Examples: Find the following sums. Assume x>0
1.
2.
3.
48 x  12 x
3
3 2x
6
 4x
4
2
1
5
3
 8x  2 x
2x
Mathematics Division, IMSP, UPLB
32
Adding and Subtracting Radicals
4.
3
1
4
x y
3
x y
2
1
3
 x xy 
3
2
 3 xy
3
3
2
x y
4
3
 3 xy y
3
2
 x 3 xy  3 y 3 xy 
1
3
2
x y
2
3
xy
3
xy
Adding and Subtracting Radicals
 x xy  3 y xy 
3
 x  3y
1
3
3
3
x 2 y2
xy
xy 
xy
3
1 3

  x  3 y   xy
xy 

 x 2 y  3xy2  1  3

 xy
xy


3
xy
3
xy
RECALL
A radical is simplified if the following hold:
1. There is no power in the radicand higher than or equal to
the index.
2. The index and the exponents in the radicand must have no
common factor.
3. There is no denominator in the radicand.
Mathematics Division, IMSP, UPLB
35
COMMON MISTAKES
1.
ab  a  b
2.
a b  ab
2
2
Mathematics Division, IMSP, UPLB
36
Exercises
A. Find the product and simplify the result:
1.
4
10 3 5 4
2 3
a b 24a b
3
2.
3 4 x 2 y 2 3 6 x 2 z 2 3 45 x 2 y 2 z
3.
4 3 5
ab
Assume
variables are
greater than
zero.
2 3
2a b
Mathematics Division, IMSP, UPLB
37
Exercises
B. Find the quotient and simplify the result:
2
1.
25 x yz
2
Assume
variables are
greater than
zero.
3 100 x 4 y 5 z 2
8 3
2.
27t s
10r
5
Mathematics Division, IMSP, UPLB
38
Exercises
B. Find the quotient and simplify the result:
3.
Assume
variables are
greater than
zero.
12 x
3
2y
2
5 34 5
4.
2 5 3 3
Mathematics Division, IMSP, UPLB
39
Exercises
C. Rationalize the denominator and simplify the result:
1.
2.
2
3
233
2  3 1
5 2
Mathematics Division, IMSP, UPLB
40
Exercises
D. Rationalize the numerator and simplify the result:
x4 2
x
1.
3
2.
xh 3 x
h
Mathematics Division, IMSP, UPLB
Assume
variables are
greater than
zero.
41
Exercises
E. Add or subtract the following radicals and simplify:
1.
75  27  12
5
3
2. 3
5
 60
3
5
Mathematics Division, IMSP, UPLB
42
Exercises
E. Add or subtract the following radicals and simplify:
3.
4.
3
3
3
3a  24a b  81a b
2
x

y
5 3
2 6
Assume
variables are
greater than
zero.
y
1 4 2 2

 x y
x
xy
Mathematics Division, IMSP, UPLB
43
Reflection
1. What is a radical?
2. What operations can be performed on
radicals? How do you do them?
3. When can you add or subtract radicals?
4. Name uses of radicals in the real world.
Mathematics Division, IMSP, UPLB
44