Chapter 11- Simplifying Radical Expressions
SPI 3102.2.1- Operate (add, subtract, multiply, divide, simplify) with radicals and radical expressions
including radicands involving rational numbers and algebraic expressions.
Section 11-6
Objective: To simplify radical (square root) expressions.
 You can use radical expressions to find the length of a throw in baseball.
Radical Expression: An expression that _____________ a radical _______sign.
Radicand: The expression ____________ the radical sign.
How to know if a radical expression is completely simplified;


The radicand has no ____________________________________ other than 1.
The radicand has no ___________________. (no fractions under the your answer CAN be
a fraction.)
There are no square roots in the ______________________.

How to break down a radical expression:
 Find the prime factorization of the numbers (and variables) in the expression.
 Group all like factors and variables as pairs of two.
 Bring out anything that can be written as a pair of two.
 Leave all items that can’t be written in a pair underneath the radical.
Example 1:
Step 1: find the ________________________.
Step 3: the two 2’s can be brought out into the
front because when you take the square root of
something squared it _________________ out.
32
22  22  2
4
2
8
2
2
4
2
2
Step 2: Group all like factors as pairs of _____.
Step 4: Leave all numbers that aren’t grouped
as pairs of two _______________ the radical.
22  22  2
Example 2:
28
You Try!:
45
I spilt the 3 y’s up into a group of two and so
there was one left over. I need everything that I
can to be in groups of two so I can cancel.
How to simplify radicals when there are variables:

48x 2 y 3
Example 1:
6
2
22  22  3  x 2  y 2  y
8
3 2 4
2

22  22  3  x 2  y 2  y
 22x y
 4xy
18a2b3 .
Example 2:
Cancel out all things with exponent of two and move
them outside the radical. All other terms stay under
the radical.
2
3  y
3y
You Try!:
50x 3 .
What about fractions?
 If you can, simplify the fractions first.
 Then, take the square root of the numerator and denominator separately.
 Break it down and simplify like we did on the first examples.
3b 2
27b 4
Example 1:

1
9b 2

1
Step 1:
9b 2
Step 2:

Step 3:
Example 2:
How did I get from the original problem to Step 1?
(Show work here)
1
32 b 2
9y 6
36y 2

1
3b
There is nothing left under the radical
because everything cancels out!
Example 3:
250q 10
5q 4
YOU TRY!!
1)
40m 3
10n 4
2)
128
25
EOC PREP:
18 x 4 y 5 in simplest radical form.
1) Write
A. 2x2y2
3y
3x2y2
6y
B.
C. 2xy
D. 3x2y2
3y 2
2y
Section 11-7
Objective: To add and subtract radical expressions, which can be used to find the perimeter of a figure.
Like radicals: Square root expressions with the same radicand.
Tell whether the following are like radicals:
1)
2 5and4 5 ____________
2) 12
2and12 5 ______________
You can only add or subtract LIKE radicals!!!
Example 1:
4 y  6 y  ______
Example 2:
6 15 
15 
15  ______
Sometimes they won’t be like terms but you will have to break down
the radicals if you can and then see if you can combine them.
108 
Example 3:
54
6
2
2
3
9
3 3
23 3 5 3  6 3 5 3
75
5
3
22  33  3 
 11 3
25
3  52
5
5 98  3 32
Example 4:
45 
YOU TRY!!! 1)
Challenge: 4 52x 
Example 5:
216t 
2) 2 3b 
180
96t
27b
117x  2 13
Section 11-8(A)
Objective: To multiply radical expressions.

Electricians can use radicals to find out how much current runs through an appliance.
Multiplying 2 radicals together:
12 
Example 1:
5
* Multiply the numbers under the radical together.
60
6
2
3
* Then break down the radicand and simplify.
10
2
(like we did in section 11-6)
5
 2235 
Example 2:
22  3  5  2 15
Example 3:





3 6




2
2 5b 
10b
YOU TRY!! 1)
2)

2



2 7




4 7x 
20x
Multiplying monomials with binomials and binomials with binomials:

You are going to multiply radical monomials and binomials just like we did with polynomials.
8( 12 
Example 1:
2)
You will need to_________________ (multiply)
the
96 
16
Break down each radical and simplify.
Can you combine these two terms? ______
Why or why not? _____________________
22  22  2  3  4
4 6 4
Example 2:


2x 
5


8 to both terms inside the parentheses.


2x 


Example 3:
2( 7  5)
YOU TRY!!!
1)


10 
5m 




4


Multiplying binomials and binomials:
2)




3
8  6




How did we multiply (2x + 5)(x – 2)?
______________________________________
Show work:
Example 1:



4






3
2



 3
2


3


8
4 3
2 3
 9
4
 3
Combine any like terms that
you may have!
84 3 2 3 
9
82 3 3
52 3
Example 2:



5






2
6



YOU TRY!!


2





3






2
5





2


EOC PREP:
1) What is the product of
A.
2 3 and 3 5 ?
2) Which expression is equivalent to
A.
B.
C.
D.
5 15
6 8
C. 5 8
D. 6 15
B.
( 5x 2 ) 4 ?
5x4
25x4
25x8
625x8
Section 11-8(B)
Objective: To learn what to do if there is a square root in the denominator.
*If there is a square root in the denominator of a fraction then the radical expression is not simplified. To
get rid of the root in the denominator we have to do something called rationalize the denominator.
Example 1:

To get rid of the root in the
denominator you have to multiply
both the top and bottom by the
6.
5
6

5
6

6
6

56 
66
30
62

30
6
Simplify the
30 if you
can.
Example 2:
Example 3:
10
13
50t
11
You Try!!!
Challenge
7
1)
2)
15
3) 
32
75k
10 2k
48z
EOC PREP:
1) If the value of the variable x is positive, what
is the sum of
3) Write
75 in simplest radical form.
7 3x and 3x ?
5 3
B. 15 5
A.
7 3x
B. 8 3x
C. 7 6 x
D. 8 6 x
A.
3 5
D. 5 15
C.
2) Which expression is equivalent to
10x
5
?
A. 2x
B.
10x
5x
2
, which expression is
3
3x 2  20 x  12
equivalent to
5
5
2
10x
D.
25
C.
4) If x ≠ 
A.
B.
C.
D.
9 x 2  12 x  4
x+6
3x + 2
–6x2 + 8x – 8
12x2 + 32x + 16
?