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Simplifying Radicals
Index
Radical
n
a
Note:
Radicand
With square roots the index is
not written
Steps for Simplifying Square Roots
1. Factor the Radicand Completely or until you find a perfect root
2. Take out perfect roots (look for pairs)
3. Everything else (no pairs) stays under the radical
Root Properties:
[1]
[2]
0 0
n
If you have an even index, you cannot take roots of
negative numbers. Roots will be positive.
 4  non - real answer
[3]
If you have an odd index, you can take the roots
of both positive and negative numbers. Roots may be
both positive and negative
3
27  3
3
 27  3
General Notes:
[1]
16  4
[2]  16  4
[3]  16  4
4 is the principal root
– 4 is the secondary root
(opposite of the principal root)
±4 indicates both primary and secondary roots
Example 1
[A]
[C]
12
24
[B]
32
[D]
120
Example 2: Simplify
[B] x10 y 6 z12
[A] x 4 y 8
[C]
x 7 y13 z12
[D] x17 y14 z 9
Example 3:
[A] 12 x 4 y10
[C]
30 x 7 y12
[B] 18 x10 y 7
[D] 49 x17 y15
Radicals CW Solutions
[1] 28 x 4 y12  2 x 2 y 6 7
[3] 20 x y
 2x y
[5] 162 x14 y10
 9 x7 y5 2
17 10
[7]
14 x 4 y12  x y
2 6
[9] 78 x17 y10
[11]
8 5
5x
14
 x8 y 5 78 x
27 x12 y 5  3x y
6 2
3y
[2]
10
256 x y
4 15
[4] 121x y
[6]
[8]
24
 16 x 5 y12
 11x2 y 7 y
45 x 4 y12  3 x y
2 6
56 x10 y17  2 x y
5 8
5
14 y
[10] 225 x 4 y13
 15 x2 y 6 y
[12] 81x14 y11
 9 x7 y 5 y
Radicals
Simplifying Cube Roots (and beyond)
1.
Factor the radicand completely
2.
Take out perfect roots (triples)
Example 1
a]
3
54
b]
3
24
Example 2
a]
3
40 x9 y11
b]
3
250 x12 y10
Example 3
[A]
[C]
Finding Roots
 16x
5
4
10 15
243a b
3
4
[B]  ( x  5)
[D]
3
64a12b 21
Example 4
[A]
Applications Using Roots
The time T in seconds that it takes a pendulum to make a
complete swing back and forth is given by the formula
below, where L is the length of the pendulum in feet and
g is the acceleration due to gravity. Find T for a 1.5 foot
pendulum. Round to the nearest 100th and g = 32 ft/sec2.
L
T  2
g
1.5
T  2
32
 1.36 seconds
Example 5
[B]
Applications Using Roots
The distance D in miles from an observer to the horizon
over flat land or water can be estimated by the formula
below, where h is the height in feet of observation. How
far is the horizon for a person whose eyes are at 6 feet?
Round to the nearest 100th.
D  1.23 h
D  1.23 6
 3.01 miles
Simplifying Radicals
1. Multiply radicand by radicand
2. If it’s not underneath the radical then do not
multiply, write together (ex: 2 3 )
Example 1
[A]
7
Multiplying Radical Expressions

2 3


5 32
[B]

15  2 5
[C]

3 5

2 3

6  3 3  5 2  15
[D]

52

3 3

15  3 5  2 3  6
Example 2 Foil
a]
( x  2)( x  4)
b]
(5  m )(7  m )
35  5 m  7 m  m
35 12 m  m
c]
(6 x  2)(3 x  4)
d]
(3 x  2)(5 x  4)
15 x  12 x  10 x  8
15x  2 x  8
Example 3
Simplify Sums / Differences
•Find common radicand
•Combine like terms
a]
2 12  3 27  3
b]
5 8  3 18  2 2
10 2  9 2  2 2
3 2
Example 4
[A]
Adding / Subtracting Roots
2 2 3 2 5 5
[B]
6 2 3 2  4 3
3 2 4 3
[C]
3 8  2 18  32
[D]
2 12  27  8
 4 3 3 3 2 2
7 32 2
SPECIAL FRACTION EXPONENT:
1
n
1
The exponent is most often used in the power of
n
monomials.
Examples: Do you notice any other type of mathematical
symbols that these special fraction exponents represent?
x 
1
3 3
5 z 
2

1
2 2

3 x 
2
1
4 2
6 x 
3
1
6 3

2 y 

9 
8
1
4 4
1
2 2


1
N
Special Fraction Exponents, a
, are more commonly known
as radicals in which the N value represents the root or index of
the radical.
Index
Radical Symbol
1
Radicals:
a  a
n
n
Radicand
Note: The square root or ½ exponent is the most common
radical and does not need to have the index written.
1
2
a  a
Steps for Simplifying Square Roots
1. Prime Factorization: Factor the Radicand Completely
2. Write the base of all perfect squares (PAIRS) outside of the
radical as product
3. Everything else (SINGLES) stays under the radical as a
product.
Operations with Rational (Fraction) Exponents
The same operations of when to multiply, add, subtract
exponents apply with rational (fraction) exponents as did
with integer (whole) exponents
•
Hint: Remember how to find common denominators and reduce.
1)
4)
2
3
x x
a
a
8
3
2)
3y 2y
6
7
2
3
7
5
5)
2x 
2
3
6
5
7
3)
6)
3
2
8z  z
5
6
5 b 
1
4
5
2
8
Radicals CW
Write in rational form.
1.
3
x5
2.
5
x7
3.
x5
4.
7
x3
Write in radical form.
5.
1
x3
6.
2
x5
7.
5
x2
8.
3
x7
Radicals (Roots) and Rational Exponent Form
Base 
Rational Exponents Property:
POWER
ROOT
m
n
x 
n

ROOT
x
m
Base
OR

POWER
ROOT
OR
( Base) POWER
 x
m
n
Example 1: Change Rational to Radical Form
A]
x
2
3
3
Example 2:
A]
3
x
5
x
2
B]
3 x 
4
5
5
C]
34 x 4
x
1
x
1

2
Change Radical to Rational Form
5
x3
B]
7
2x
9
1
7
2 x
9
7
C]
6x  6 x 
5
5
2
Radicals Classwork
# 1 – 4: Write in rational form.
1.
3
x
5
2.
5
8x
7
3.
x5 y6
4.
7
4a 8b 3
#5 – 8: Write in radical form.
5.
x
1
3
6.
7 x 
2
5
7.
x
5

2
8.
x
3
7
Radicals Classwork #2
Determine if each pair are equivalent statements or not.
1.
3.
3
4
x and
6
4
5.
 x
x and
5
3
x and
4
3
2
x
x
5
3
3
2.
5
x 3 and
 x  and
4.
6.
6
7
x
x
5
3
18
6
and
14
x
x
3
9
12
Simplifying Rational Exponents
• Apply normal operations with exponents.
• Convert to radical form.
• Simplify the radical expression based on the index and radicand.
1.
5.
27
a
1
3
8

5
2.
8
6. 9b
2
3

2
4 3
3.
5
x2
7. 8x

5
5 7
4.
4 x 
8. ab c
3
7

4
2 3 9
Radicals Classwork #3
Simplify the following expressions into simplest radical form
1.
x
x
4.
8
9
2.
5
9
3
5
4 4
a
a
4
5
5.
2
3
3.
12

3
5
1
6
5  3 52
6.
1
3
x x
3
4
Change of Base (Index or Root)
• Write the radicand in prime factorization form
• REDUCE the fractions of Rational Exponents to rewrite radicals.
1.
6
27
3.
10
15
x
4
2.
4.
8
4
64
3.
3
3.
4
b
b
6
6
Change of Base Practice Problems
1.
4. 6
6
25
16a
2. 6
8
27x
5. 10
3. 4
3
32a
5
6. 4
81a
2
10
4 2
36 x y z
Radicals
Radical Equation
Equation with a variable under the radical sign
Extraneous Solutions
Extra solutions that do not satisfy equation
Radical Equation Steps
[1] Isolate the radical term (if two, the more complex)
[2]
Square, Cube, Fourth, etc. Both Sides
[3]
Solve and check for extraneous solutions
Example 1
[A]
Solving Radical Equations Algebraically
x 1  2  4
[B]
y  2 1  5
y2 6
y  2  36
y  38
Example 1
[C]
x  1  5  15
x  1  10
x  1  100
x  101
[D]
y  3 1  7
y 3 8
y  3  64
y  61
Radicals CW
Solve Algebraically.
9.
x  5  12  4
10.
x  5  16
x  5  256
x  251
11.
2 x  2  11  17
2x  2  6
2 x  2  36
2 x  38
x  19
12.
x  6  10  4
x  6  14
x  6  196
x  190
3 5x  1  11  17
3 5x  1  6
5x  1  2
5x  1  4
5x  5
x 1
Radicals CW
Solve Algebraically.
13.
4 2x  1  5  9
4 2x  1  4
2x  1  1
2x  1  1
2x  0
x0
14.
7 4 x  1  2  16
7 4 x  1  14
4x  1  2
4x  1  4
4x  3
3
x
4
Radicals CW
Solve Algebraically.
15.
x 5  x 7  4
No Solution
16.
x  21  1  x  12
x=4
Example 2
[A]
Solving Graphically
3x  1  5 x  1
[B]
2x  3  3  2x
x=½
Example 2
[C]
Continued
y  21 1  y  12
Y=4
[D]
x  1  x  6 1
x=3
Example 3
[A]
No Solutions
x  15  3  x
[B]
3x  2  x  4
x=Ø
Example 4
[A]
Misc. Equations
4  x  x2  8
[B]
3x  7  x  3
x = -1, -2
x=3