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Algebra 2
Notes 7-1
Name________________________
Date____________Period______
7-1: Roots and Radical Expressions
Def: The nth root of an equation: For any real numbers a and b , and any positive integer n , if
a n  b then a is the nth root of b .
Type of Number
Number of Real nth Roots
When n Is Even
Number Of Real nth Roots
When n is Odd
Positive
0
Negative
Radical Expressions Vocab
index

na
radicand

The principal root is the positive root of a number that has a positive and negative
root.
Property: nth root of a n , a  0

For any negative real number a ,
n
a n  a when n is even.
Ex: Simplify each radical expression
4x 6
3
a 3b 6
4x 2 y 4
3
27c 6
4
4
x 4y 8
x 8 y 12
HW: ______________________________________________
Brashear Alg 2 Ch 7 notes (3/23/09)
page 1 of 8
Algebra 2
Notes 7-2
Name________________________
Date____________Period______
7-2: Multiplying and Dividing Radical Expressions
Multiplying Radical Expressions

If
n
a and
n
b are real numbers, then n a
n
b  n ab .
Ex: Multiply and simplify answer if necessary.
3 12


3
3
3

4

4
4
3x
40n 2
3
2n 3
9


3
Simplify 3 54x 2 y 3 3 5x 3 y 4 .
Assume all variables are
positive.
4
24x 2
Dividing Radical Expressions

If
n
a and
n
Brashear Alg 2 Ch 7 notes (3/23/09)
b are real numbers and b  0 , then
n
a na
.

n
b
b
page 2 of 8
Ex: Divide. Assume all variables are positive.
3
32
3
4
3
162 x 5
3
3x 2
4
1024x 15
4
4x
Rationalizing the denominator
 To be in simplest form, a radical expression should have all perfect roots taken out of
the radicand, and it should not have a radical in the denominator.
Ex: Rationalize the denominator of each expression. Assume all variables are positive.
2
3
7
5
x3
5xy
3
2
3x
2x 3
3
10xy
3
HW: _________________________________________________
Brashear Alg 2 Ch 7 notes (3/23/09)
page 3 of 8
4
6x
Algebra 2
Notes 7-3
Name________________________
Date____________Period______
7-3: Binomial Radical Expressions
Like Radicals are radicals that have the same index and the same radicand. If radical terms are
“like”, we can add or subtract them.
Ex: Add or subtract if possible. It may be necessary to simplify the radicals first to see if you have like
terms.:
5 75  2 12
2 3  3 27
53 x  33 x
50  4 32  3 12
5 3  7 12  3 75
Multiplying and Dividing Binomial Radical Expressions
Ex:
 2  2 5 6  2 5 
 3  2 5  2  4 5 
7  6 8 
2
Brashear Alg 2 Ch 7 notes (3/23/09)

5 6

5 6

page 4 of 8
Rationalizing Binomial Denominators

Multiply by a fraction of
conjugate of denominator
. Simplify.
conjugate of denominator
3 5
5 7
1 5
4 3
HW: _________________________________________________________
Brashear Alg 2 Ch 7 notes (3/23/09)
page 5 of 8
Algebra 2
Notes 7-4
Name________________________
Date____________Period______
7-4: Rational Exponents
Another way to write a radical expression is with a rational (fraction) exponent. See the examples
below:
1
25  5 2
1
3
27  27 3
4
16  16 4
1
Try these. Simplify each expression:
1
1
125 3
1
1
1
1
1
1
10 3 100 3
52 52
1
22 82
22 22
What to do when the numerator of a rational exponent is other than 1:
 If the n th root of a is a real number and m is an integer, then
1
an na
and a
m
n
 n am 
 
n
a
m
. If m is a negative, a  0 .
Ex: Convert from radical to rational exponent form, or vice versa.
3
x5
Brashear Alg 2 Ch 7 notes (3/23/09)
y 2.5
z3
 b
5
2
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Ex: Simplify:
 32 
3
5
6
15
25
5
 81  2
32 5
 8x 
4
3.5

1
3

3
2
 16 y 
8

7
 43
 1
x x 8  x 6


HW: _________________________________________________________
Brashear Alg 2 Ch 7 notes (3/23/09)
page 7 of 8
3
4
Algebra 2
Notes 7-5
Name________________________
Date____________Period______
7-5: Solving Radical Equations


Isolate the radical expression/rational exponent on one side of the equation.
Raise both sides of the equation to the same power. (If trying to undo a square root,
square both sides. If trying to undo a cube root, cube both sides. If solving for an x
raised to the

2
3
power, raise both sides to the
3
, etc.)
2
Check for extraneous roots! Plug your answer(s) back into the original equation to
make sure they work!
Ex: Solve
2  3x  2  6
5x  1  6  0
2
3
2  x  2  3  50
 2x  1 
0.5
  3x  4 
3  x  3  2  54
0.25
0
3x  2  2 x  7  0
HW: ______________________________________________________________
Brashear Alg 2 Ch 7 notes (3/23/09)
page 8 of 8