Ch 10: Radical Expressions and Equations
10-1 Simplifying Radicals
10-2 The Pythagorean Theorem
10-3 Operations With Radical Expressions
10-4 Solving Radical Equations
10-5 Graphing Square Root Functions
10-1 Simplifying Radicals:
Vocabulary:
 Radical expressions
 Perfect Square
 Rationalize
Focused Learning Target:
 Simplify Radicals
 Rationalize denominators
Radical Expression: a radical expression is an expression containing a square root.
Perfect Square: a number which is a square of an integer. The square of a variable expression
can also be considered a perfect square.
Integer
Perfect
Variable
Perfect
Square
expression
Square
x
0
0
x2
1
1
2
4
3
9
4
16
x2
x3
x4
x5
x6
5
x4
x6
x8
6
Multiplication Property of Square Roots: For every number a  0, b  0, ab  a b .
Example: 72  36 2  6 2
Simplify Radicals by removing perfect square factors:
simplify : 45
45  9 5
45  3 5
Factor 45 so that one of the factors is a perfect square.
Rewrite using the multiplication property of Square Roots.
Simplify by taking the square root of the perfect square (9).
Check to see if any perfect square factors remain.
You Try: simplify:
8
18
50
225
Simplify Radicals with variable factors:
50 x 5
Factor out the perfect squares and rewrite using the
multiplication property of square roots.
25  2 x x
4
Simplify by taking the square root of the perfect squares.
5x2 2x
Check to see if any perfect square factors remain.
You Try: simplify
9x 4
12x 2 y
18x3
100x7
75x 5 y 2
Simplifying radicals by multiplying first: simplify
5 15
75
25  3
5 3
You try:
7 14
6 3
10 5
Division Property of Square Roots: for every number a  0, & b  0,
a
a

. Example:
b
b
16
16 4


25
25 5
You Try: Simplify
9
36
49
100
64
25
Simplifying radicals by dividing:
45
 9 3
5
39 x 3

52 x 5
13  3 x 3
13  4 x 3  x 2
3
3

2
4x
4x2
3
2x
You Try: Simplify
90
5
48
75
27 x3
3x
Rationalizing a denominator: to transform a fraction with a radical in the denominator, into an
equivalent fraction with no square roots in the denominator. Multiply the numerator and
denominator by the same radical that is in the denominator.
Ex:
10
5
3 5 3 5 3



54
3
3 3
9
5 2
27  2
5
93
5
3

3 3 3
15
33
15
9
Reduce the fraction inside the
radical.
Use the multiplication property of
square roots to simplify the
denominator.
Rationalize the denominator and
simplify
You try: Rationalize each denominator
8
7
10-2 The Pythagorean Theorem:
Vocabulary:
 Hypotenuse
 Leg
 Pythagorean Theorem
18
14
3
11
Objective:
 Use the Pythagorean Theorem to determine the
lengths of the sides of a right triangle.
 Use the Converse of the Pythagorean Theorem
to determine if a triangle is a right triangle.
In a right triangle, the side opposite the right
angle is called the hypotenuse. It is always the
longest side. Both of the remaining sides are
called legs.
Hypotenuse
Leg
Leg
The Pythagorean Theorem: describes the
relationship between the hypotenuse and the
legs. In any right triangle, the sum of the
squares of the lengths of the legs is equal to
the square of the length of the hypotenuse:
a 2  b2  c2
C
a
b
Using the Pythagorean theorem:
Find the length of the hypotenuse:
C
6 in
8 in
a  6 in, b  8 in, c  c
a 2  b2  c2
1. Identify the legs and the hypotenuse and
substitute into the Pythagorean theorem.
6 2  82  c 2
36  64  c 2
2. simplify
100  c 2
100  c 2
3. Take the square root of both sides.
10  c
Applications (word problems)
When solving word problems, using the Pythagorean Theorem, be sure to draw the picture. A
Pretty picture is not important, but finding and drawing the triangle is very important.
Ex: Firemen entering a burning building place a 30 ft. ladder 16 feet from the building. How
high is the top of the ladder?
1. Draw the picture
2. Find the right triangle and label the sides.
3. Use the Pythagorean Theorem to solve the
problem.
You Try:
The sail on a sailboat is a right triangle. If the
longest edge is 25 ft. and the base of the
triangle is 15 feet, how high is the sail?
The sail on a sailboat is a right triangle. The
longest edge is 13 ft. and the base of the
triangle is 5 feet. If the base of the sail is 3
feet above the deck, how high above the deck
is the top of the sail?
Converse of the Pythagorean Theorem: If a triangle has side lengths of a, b & c so that
a 2  b 2  c 2 , then the triangle is a right triangle with hypotenuse c.
Ex: Determine if the following sides of a triangle can be the sides of a right triangle.
3 in, 7 in., 10 in.
1. Define the sides. The longest side must be the
a = 3, b = 7, c = 10
hypotenuse, c.
32  7 2 ? 102
9  49  100
2. Plug the values into the Pythagorean Theorem
and evaluate.
3. If it is equality, than it is also a right triangle. If
not, it is not a right triangle.
10-3 Operations with Radical Expressions:
Vocabulary:
 Like radicals
 Unlike radicals
 conjugates
Objective:
 simplify radical expressions
Tool Box:
 Distributive Property
 FOIL
 Difference of Squares
Like radicals: like radicals have the same radicand.
Unlike radicals: unlike radicals have different radicands.
Like Radicals:
5, 6 5
2x , y 2x
Unlike Radicals:
5, 5 6
x , 3 xy
Like radicals can be combined by adding/subtracting the “coefficient” of the radical. Unlike
radicals cannot be combined.
Ex: Simplify
5 6 5 
a x b x 
13 x  4 x 
x y 
You Try: Simplify
8 2 6 2 
x x
3 2 
Simplify radical expressions using the distributive property:
5

15  3

5 15  3 5
75  3 5
25 3  3 5
5 3 3 5
1. Multiply using the distributive property.
2. Use the multiplication property of square roots to simplify.
3. Check to see if any of the radicals can be simplified using
perfect squares.
4. Finish simplifying.
You Try: Simplify
3

6 7


5 2  10
Simplify radical expressions using FOIL:

2x


6 x  11

3 6

32 6

3 32 3 6  3 6 2 6 6
1. Multiply using FOIL.
9  2 18  18  2 36
2. Check to see if any of the radicals can be simplified
using perfect squares and combine like terms.
3  18  2  6
9 2 9
3. Use the multiplication property of square roots to
simplify, and combine like radicals.
3 2 9
4. Check again to see if there are any like radicals or like
terms to combine.
5. Finish simplifying.
You Try:
2
6 3 3

6 5 3


2 3

2
10-4 Solving Radical Equations:
Vocabulary:
 Radical Equations
 Extraneous Solution
Tool Box:
 Simplifying radical expressions
 Solving equations (isolate a variable)
Solving by Isolating the Radical:

6 5

6 5

Objective:
 Solve Radical Expressions Dude!
Ex: Solve for x

2 x  6  8  16
8 8
1. Isolate the Radical.

 82
2. Square both sides.
2x  6
6
 64
6
3. Solve for the variable.
2x
2
x

2x  6
2
70
2
 35
4. Check your answer.
2 x  6  8  16
2  35   6  8  16
64  8  16
8  8  16
You try: solve for x
x 7
x  4  11
Solving With Radical Expressions on Both Sides:
x  5  8  10
Solve for x:
2 x  18  4 x  6
2
2 x  18  4 x  18
2 x  18  4 x  6
6
6
2
1. Square both sides.
2. Solve for the variable.
2 x  24  4 x
2 x
 2x
24 2 x

2
2
12  x
2 x  18  4 x  6
3. Check your answer.
2 12   18  4 12   6
24  18  48  6
42  42
You try: solve for x
5x  3  6 x  2
Identifying Extraneous Solutions:
x  3  3 x  11
7 x  5  9 x  11
If the work was done correctly, but one of the solutions does not make sense or is undefined,
than the answer is an extraneous solution.
Ex: solve for x
x
 x2
x2
 x2
2
1. solve the radical equation
x2
 x2
x2 x2
x2  x  2  0
 x  2  x  1  0
x20
x
2
x 1  0
x
x x2
 2   2  2
 1
x x2
1 
 1  2
2 4
2. Check
3. Because the square root of a number is
never negative, -1 is an extraneous solution.
(note: it is possible to have a negative answer,
but not if it leads to the square root being
negative.)
You try: solve for x
x  x6
10-5 Graphing Square Root Functions:
x  3x  2
Vocabulary:
 Square Root Function
Objective:
 Graph the Square Root function
1. Start by setting the expression inside the radical = 0. Use the answer for your first value of x.
2. Make a table with 3 or more ordered pairs and draw a smooth curve. You can estimate the
square root of any values that are not perfect squares or choose only perfect squares. (all
values of x must be greater than the first value chosen.
Example:
y
Graph:
4
The least value of x is 0.
plug in and make your table of values.
x
y
x




2
x
-2
2
4
6
y
x




8
-2
-4
Example:
Graph: y 
4
x4
y  x4
x40
y
4 4
2
x4
x
-2
2
-2
-4
4
6
8 10 12
1st value for x = 4
x
y
x4
4
 4  4



4
4
4
You Try: Graph y 
4
x2
Least value for x =
x
x2
y
y
2
x
-2
2
4
6
8 10 12
-2
-4
You Try: y 
4
x 4
Least value for x = 0
x
y
x 4
y
2
x
-2
2
-2
-4
4
6
8 10 12