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Module 3 Review Lessons
What is a square root?
A square root is number that is multiplied by itself to equal a quantity. A square root is in the family of
“radicals”.
is called a “radical” or “root” sign, but in this case we call it a “square root”.
Example:
25 means “what number is multiplied by itself to equal 25?” and 25 is called the “radicand”
How are squaring and square root related?
Finding a square root is the inverse operation of squaring a number, like inverses + and – or inverses x and ÷.
Square
Square Roots
32  3  3  9
9  33  3
Three squared equals 3 times 3, which equals 9.
3 units
The square root of 9 equals the square root of 3x3 = 3.
3 units
3 units
3 units
32 = 9
square
units
√9 square units
= √32
= 3 units
What is a perfect square?
A perfect square is a number that can be expressed as the product of two equal integers. For example: x2 is
the perfect square for the quantity x. Here are the perfect squares of the first ten natural numbers 1 - 12:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144…
Square Root Property
For any real numbers a and b, if a2 = b, then a is a square root of b and
b a.
Every non-zero real number has both a positive and negative square root possible.
9  3  3  3 or
9  3  3   3 , so
9  3 → 2 possible real square roots
0  0  0  0 → 1 possible real square root
9  3  3 → no possible real square root
When you are given a square root sign, you must give the “principal square root” answer, which will never be
the negative square root. You will only have negative answers if given the negative sign.
Examples:
a. 121  11
b.
36  6
c.
1 1
d.  100   10
e.  81   9
Simplifying Square Roots
A square root is simplified when there are no longer perfect square number factors or variable factors under
the square root sign other than 1. A square root on the bottom of a fraction is not considered simplified.
3
2
4 13
Examples: Simplified 7
Not Simplified
40
3 68x2
6n 21
2
3
We will now use the square root property to simplify square roots. There are 2 different methods.
Method 1: Factor the radicand into perfect squares. Remove every perfect square by taking its square root.
Examples- Simplify each square root
a. 90  9  10  3 10
b.  252   36  7  6 7
Method 2: Factor the radicand into prime numbers (called prime factorization- you may want to make a
factor tree). Remove every perfect square by taking its square root (2 of the same factor comes out as 1).
Examples- Simplify each square root
a. 90  2  3  3  5  3 2  5  3 10
b.  252   2  2  3  3  7  2  3 7  6 7
Simplify with Variables
Simplifying with variables If x  x  x 2 , then the square root of x2 is x:
x2  x  x  x .
Examples- Simplify each square root
8 99 x 5 y 3 z 2 
x3 y 4 
a.
xxx y y y y 
b.
x y y x 
8 3  3 11  x  x  x  x  x  y  y  y  z  z 
8  3  x  x  y  z 11  x  y 
24 x 2 yz 11xy
xy 2 x
Simplified Radical Expressions
 No radicands have perfect square factors other than 1.
 No radicands contain fractions.
 No radicals appear in the denominator of a fraction.
Multiplying Square Roots
Multiply
To multiply radicals, you can either factor both radicals first and then multiply or multiply first, then simplify.
Multiplication Property of Square Roots:
In this case, both methods are fairly easy:
Example:
a  b  ab
9  4  32  6
and
15x 3y  15xy
Method 1- Multiply first
15 x 3 y  15 xy
Method 2- Factor first
15 x 3 y  15 xy
 225 x 4 y 2
 15 15  x  x  x  x  y  y
 15  15  x 2  x 2  y 2
 15  x  x  y
 15 x 2 y
 15 x 2 y
9  4  36  6
Dividing Square Roots
Divide
To divide radicals, you can factor both radicals first and then divide OR divide first, then simplify.
Division Property of Square Roots:
In this case, both methods are fairly easy:
Example 1:
40 x 2
4 10  x 2 2 x 10


 x 10
2
4
4
16
4
OR

4
2
2
a
a

b
b
and
16
4
 16
4
 4 2
40 x 2
40 x 2

 10 x 2  x 10
4
4
In this case, the second method is probably easier.
Example 2:
Example 3:
Example 4:
100
100 10
But in this case, it is impossible to divide them first.


81
9
81

90x 18
 45x 17  3  3  5  x 8  x 8  x  3x 8 5x
2x

6x 8 y 9

5x 2 y 4
6x 6 y 5

5
x 3 y 2 6y
6x 6 y 5

5
5
This last example is not completely simplified since it has a square root in the denominator.
Square roots are “irrational” numbers and they cannot be in the denominator. You must
rationalize the denominator (that means make it a rational number).
Rationalize the Denominator- Dividing by a Square Root that is not a perfect square
Rationalize the Denominator
Notice that after simplifying the radical, we still have a square root in the denominator. We cannot leave a
square root on the bottom of a fraction, so we have to find a way to get rid of the radical. This is called
rationalizing the denominator (that means making it a rational number).
When the denominator is a monomial (one term), multiply both the numerator and the denominator by the
square root on the bottom.
If b is not a perfect square and not zero:
a
a
b a b



b
b
b b
or
a
a b
ab



b
b
b b
Examples:
6
3
7
2

a.


7
2

2
2
14

Multiply top/bottom by √2
4
14
2
b.
6
3

3
3

6 3
9

6 3
2 3
3
Simplify after you divide
3  2 and 3  2 are called “conjugates”. If one of them is in the denominator, we multiply by its conjugate
to rationalize it.
c. In this one, we multiply top and bottom by the conjugate 2+√5. We use FOIL on the bottom, then
simplify.
3
2 5


3 2 5

 2  5  2  5 

2 3  3  5
2 3  15 2 3  15 2 3  15



 2 3  15
45
1
22  2 5  2 5  5  5
4  25
Adding and Subtracting Square Roots
Adding and subtracting radicals is similar to adding and subtracting polynomial terms. You cannot combine 4x
and 6y because they are not like terms, and you cannot combine radicals unless they are “like radicals”.
You can simplify these a. 2x + 3x = 5x
b.
2 5 3 5  5 5
c. 17  7 17  6 17
DO NOT ADD THE NUMBERS UNDER THE RADICAL!
These are simplified
a. 2x + 3n
b. 2 5  5 2
c. 11  9 6
These do not look like “like radicals”, but if you simplify the square roots then you can add or subtract them.
Simplify 3  12
Recall the 12  2  2  3  4  3  2 3
Example:
Simplify
Note: Simplify the square roots before you add!

 4 7  83 7

63  3  3  7  3 7
 4 7  24 7
 28 7
Example: A garden has width
and length
. What is the perimeter of the garden in simplest radical
form? Recall that perimeter is the distance around an object. (P = 2L + 2W is the perimeter of a rectangle)
P  2( 13 )  2(7 13 )
P  2 13  14 13
P  16 13
Use the formula for perimeter and substitute the value of the width and length. Simplify.
Polynomials and Quadratic Equations
Polynomial- an algebraic expression with constants, variables, and exponents combined using addition,
subtraction, or multiplication. It may have one or more terms, but they cannot be infinite. Exponents can only
be 0, 1, 2, 3, etc.
3
Polynomials 3x2-4x, -1,
3-x,
5x2y +7xy+y2
NOT Polynomials , 5 x 2 y, 2 x
x
Polynomials can be named two different ways: number of terms OR degree/highest power
By the number of terms:
Monomial- one term 4x2y ,
Trinomial- three terms
Binomial – two terms 4x2y – 3x, x + 2
-3x
5x2y +7xy+y2 ,
By the degree/highest power:
Constant- degree of 0
-1,
Quadratic- degree of 2
9
6a2-4a-1,
Forms of a Quadratic Equation
Examples
6a3-4a-1
Linear= degree of 1
a ,
-3x + 7
3x2-4x
Standard Form:
ax2 + bx + c = 0
2x2 - 5x + 9 = 0
Factored Form:
(x – p)(x – q) = 0
(x + 11) (x – 9) = 0
**Recall how to multiply using laws of exponents
EX: (5x2y)(3xy3) = 15x3y4
You multiply the coefficients, multiply the x’s, multiply the y’s
EX:
(½an)(8n) = 4an2
You multiply the coefficients, multiply the n’s, bring over the a.
**Recall how to multiply two binomials: You multiply using “FOIL” (First, Outer, Inner, Last) and then combine
the like terms. FOIL is an easy way to remember to use the Distributive Property to multiply each term.
EX: (x – 2) (x + 5) = x2 + 5x – 2x – 10 = x2 + 3x -10
EX: (n + 8)(n + 6) = n2 + 6n + 8n +48 = n2 + 14n + 48
Now we will look at a pattern for multiplying binomials using a “Box” method. It is the same thing as FOIL, and
we will look at the pattern of the coefficient and constants in the answer.
FOIL
x
-2
x
x
2
-2x
+5
5x
-10
FOIL
FOIL
F
I
O
L
n
2
+8
n
n
+8n
+6
+6n
+48
Multiply 2 Binomials Review- FOIL and BOX methods
These are examples of multiplying two binomials using FOIL in the Box format.
Binomial Factors
Box/FOIL Work
Product of Binomials
X Factors
( x  8)( x  4)
( x  8)
2
( x  8)( x  8)
( x  8)( x  4)
( x  8)( x  4)
( x  8)( x  4)
x2
4x
32
x 2  4 x  32
x2
8x
64
x 2  16 x  64
8x
x2
8x
8x 64
4x
x2
8x 32
x2
4x
8x
x 16x  64
x2
4x
(3n + 1)(2n + 5)
8x
6n 2
15n
2
x 2  64
x 2  12 x  32
x 2  4 x  32
8x
32
x 2  12 x  32
2n
6n 2  17 n  5
5
Looking at the X Factors for the Binomials
Now let’s look at the numbers that we placed inside the X. How do they relate to the problem?
Binomial Factors
Box/FOIL Work
Product of Binomials
X Factors
( x  8)( x  4)
( x  8)
2
( x  8)( x  8)
( x  8)( x  4)
x2
8x
4x
x2 16x  64
x  4 x  32
x2
8x
64
x  16 x  64
8x
x2
8x
8x 64
4x
x2
8x x  4x  32
2
( x  8)( x  4)
( x  8)( x  4)
x2
4x
8x
x 16x  64
x
2
4x
(3n + 1)(2n + 5)
6n 2
15n
2
2
2
x  64
2
x 2  12 x  32
x 2  12 x  32
8x
32
x 2  12 x  32
2n
6n 2  17 n  5
5
-32
-8
4
-4
64
8
-8
8
16
-64
-8
8
0
32
-12
-4
-32
8
-4
4
8
32
12
4
?
Patterns for Multiplying Binomials using the Box and X Factors
Let’s look at the patterns for the multiplication of binomials. Here are some important patterns.
( x  p )( x  q )
Quadratic in factored form
x  bx  c
x
-q
-p
F
I
O
L
Multiply=
2
Quadratic in standard form
x
p
c
q
Add =
b
You should notice that we placed the numbers from each factor on the left and right side of
the X. How does the bottom number relate to those two numbers? It is the sum (b)! What
about the top number? It is the product (c)!
What if there is a coefficient in front of the x?
(3n + 1)(2n + 5) =
152
3 5
2 1
15+2