Simplifying Radicals

Simplifying Radicals
Objectives:
1. To use properties to
simplify square
roots
Assignment:
• P. 874: 1-28 S
• P. 883: 1-12 S
• P. 883: 25-39 S
Objective 1
You will be able to
use properties to
simplify square
roots
Example 1
Solve x2 – 4 = 0 by
factoring.
Example 1
Solve x2 – 4 = 0 by
factoring.
Notice that we could
have solved this
equation by taking a
square root:
2
x 40
x2  4
x2  4
But where’s the other solution x = −2?
x2
Square Roots
The number r is a square root of x if r2 = x.
• This is usually written 𝑥 = 𝑟
Radicand
Radical
Square Roots
The number r is a square root of x if r2 = x.
• This is usually written 𝑥 = 𝑟
Any positive
number has two
real square roots,
one positive and
one negative, 𝑥
and − 𝑥
4 = 2 and −2,
since 22 = 4 and
(−2)2 = 4
The positive square
root is considered the
principal square root
Example 2
Use a calculator to evaluate the following:
1. 3  2
2. 6
3. 3  2
4. 3 / 2
Example 3
Use a calculator to evaluate the following:
1. 3  2
2. 5
3. 3  2
4. 1
Properties of Square Roots
Properties of Square Roots (a, b > 0)
Product Property
ab  a  b
18  9  2  3 2
Quotient Property
a
a

b
b
2
2
2


25
5
25
Simplifying Square Root
The properties of square roots allow us to
simplify radical expressions.
A radical expression is in simplest form
when:
1. The radicand
has no nontrivial perfectsquare factor
2. There’s no
radical in the
denominator
Example 4
Write the first 20 terms of the following
sequence:
1, 4, 9, 16, …
x
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
2
0
x2
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
289
324
361
400
These numbers are called the
Perfect Squares.
Simplest Radical Form
Like the number
3/6, 75 is not in
its simplest form.
Also, the
process of
simplification for
both numbers
involves factors.
• Method 1: Factoring
out a perfect square.
75 
25 3 
25  3 
5 3
Simplest Radical Form
In the second
method, pairs of
factors come out
of the radical as
single factors,
but single
factors stay
within the
radical.
• Method 2: Making a
factor tree.
75 
25
5
3
5
5 3
Simplest Radical Form
This method
works because
pairs of factors
are really perfect
squares. So 5·5
is 52, the square
root of which is
5.
• Method 2: Making a
factor tree.
75 
25
5
3
5
5 3
Investigation 1
Express each square root in its simplest form
by factoring out a perfect square or by
using a factor tree.
12
48
18
60
24
75
32
83
40
300x
3
Example 5a
Simplify the expression.
27
10  15
9
64
11
25
Example 5b
Simplify the expression.
98
8  28
15
4
36
49
Example 6
Simplify the expression.
1. 3 24
4
2. 162
Example 6
Evaluate, and then classify the product.
1. (√5)(√5) =
2. (2 + √5)(2 – √5) =
Conjugates are Magic!
The radical expressions 𝑎 + 𝑏 and 𝑎 − 𝑏
are called conjugates.
The product of two conjugates is always a
rational number
𝑎+ 𝑏 𝑎− 𝑏 =
𝑎2
2
− 𝑏 =
𝑎2 − 𝑏
Example 7
Identify the conjugate of each of the following
radical expressions:
1. √7
2. 5 – √11
3. √13 + 9
Example 8
Recall that a radical expression is not in
simplest form if it has a radical in the
denominator.
How could we use conjugates to get rid of
any damnable denominator-bound
radicals?
Rationalizing the Denominator
We can use conjugates to get rid of radicals
in the denominator:
The process of multiplying the top and bottom of a
radical expression by the conjugate of the
denominator is called rationalizing the
denominator.


5 1 3
1 3
5
5  5 3 5  5 3




2
2
1 3 1 3 1 3 1 3

Fancy One


Example 9a
Simplify the expression.
6
5
6
7 5
17
12
1
9 7
Example 9b
Simplify the expression.
19
21
9
8
2
4  11
4
8 3
Solving Quadratics
If a quadratic equation has no linear term, you can
use square roots to solve it.
•
By definition, if
x2 = c, then 𝑥 =
𝑐 and 𝑥 = − 𝑐,
usually written
𝑥=± 𝑐
•
You would only
solve a quadratic
by finding a square
root if it is of the
form
ax2 = c
In this lesson, c >
0, but that does not
have to be true
Solving Quadratics
If a quadratic equation has no linear term, you can
use square roots to solve it.
By definition, if
x2 = c, then 𝑥 =
𝑐 and 𝑥 = − 𝑐,
usually written
𝑥=± 𝑐
To solve a quadratic
equation using square
roots:
1.Isolate
the
squared
term
2.Take the
square
root of
both
sides
Example 10a
Solve 2x2 – 15 = 35 for x.
Example 10b
Solve for x.
1
2
 x  4   11
3
Simplifying Radicals
Objectives:
1. To use properties to
simplify square
roots
Assignment:
• P. 874: 1-28 S
• P. 883: 1-12 S
• P. 883: 25-39 S