Martin-Gay, Developmental Mathematics 1
Warm-Up #5
1. Find the product of ab. .
2. Simplify
3. Estimate: What is
Martin-Gay, Developmental Mathematics 2
Homework
Advanced: Simplifying Radical Worksheet
Page 1. #1-6
Page 2. #1-6
Regular: Simplifying Radical Worksheet
Page 1. #1-4
Page 2. #1-4
Introduction to Radicals
Martin-Gay, Developmental Mathematics 4
The principal (positive) square root is noted as
a
The negative square root is noted as
a
Principal Square Roots
Martin-Gay, Developmental Mathematics 5
Perfect Squares
1
4
916
253649
64
81
100121
144169196
225
256
324
400
625
289
Martin-Gay, Developmental Mathematics 6
16
25
100
144
= 4 or -4
= 5 or -5
= 10 or -10
= 12 or -12
Martin-Gay, Developmental Mathematics 7
20
32
75
40
= =
=
=
5*4
2*16
3*25
10*4
=
=
=
=
52
24
35
102
Perfect Square Factor * Other Factor
LE
AV
E I
N R
AD
ICA
L F
OR
M
Martin-Gay, Developmental Mathematics 8
The cube root of a real number a
abba 33 ifonly
Example:
Cube Roots
8)2)(2)(2(2 because 28 33
Martin-Gay, Developmental Mathematics 9
Cube Roots
3 27
A cube root of any positive number is positive.
Examples:
3 5
43
125
64
3 8 2
A cube root of any negative number is negative.
3 a
15.1 – Introduction to Radicals
3 27 3 3 8 2
Martin-Gay, Developmental Mathematics 10
3 27 3
3 68x 22x
Cube Roots
Example
Simplifying Radicals
Martin-Gay, Developmental Mathematics 12
baab
0b if b
a
b
a
a bIf and are real numbers,
Product Rule for Radicals
Martin-Gay, Developmental Mathematics 13
Simplify the following radical expressions.
40 104 102
16
5 16
5
4
5
15 No perfect square factor, so the radical is already simplified.
Simplifying Radicals
Example
Martin-Gay, Developmental Mathematics 14
Simplify the following radical expressions.
7x xx6 xx6 xx3
16
20
x
16
20
x
8
54
x 8
52
x
Simplifying Radicals
Example
Martin-Gay, Developmental Mathematics 15
nnn baab
0 if n
n
n
n bb
a
b
a
n a n bIf and are real numbers,
Quotient Rule for Radicals
Martin-Gay, Developmental Mathematics 16
Simplify the following radical expressions.
3 16 3 28 33 28 3 2 2
3
64
3 3
3
64
3
4
33
Simplifying Radicals
Example
Adding and Subtracting Radicals
Martin-Gay, Developmental Mathematics 18
Sums and Differences
Rules in the previous section allowed us to split radicals that had a radicand which was a product or a quotient.
We can NOT split sums or differences.
baba
baba
Martin-Gay, Developmental Mathematics 19
What is combining “like terms”?
Similarly, we can work with the concept of “like” radicals to combine radicals with the same radicand.
Like Radicals
Martin-Gay, Developmental Mathematics 20
373 38
24210 26
3 2 42 Can not simplify
35 Can not simplify
Adding and Subtracting Radical Expressions
Example
Martin-Gay, Developmental Mathematics 21
Simplify the following radical expression. 331275
3334325
3334325
333235
3325 36
Example
Adding and Subtracting Radical Expressions
Martin-Gay, Developmental Mathematics 22
Simplify the following radical expression.
91464 33
9144 3 3 145
Example
Adding and Subtracting Radical Expressions
Martin-Gay, Developmental Mathematics 23
Simplify the following radical expression. Assume that variables represent positive real numbers.
xxx 5453 3 xxxx 5593 2
xxxx 5593 2
xxxx 5533
xxxx 559
xxx 59 xx 510
Example
Adding and Subtracting Radical Expressions
Multiplying and Dividing Radicals
Martin-Gay, Developmental Mathematics 25
nnn abba
0 if b b
a
b
an
n
n
n a n bIf and are real numbers,
Multiplying and Dividing Radical Expressions
Martin-Gay, Developmental Mathematics 26
Simplify the following radical expressions.
xy 53 xy15
23
67
ba
ba
23
67
ba
ba44ba 22ba
Multiplying and Dividing Radical Expressions
Example
Martin-Gay, Developmental Mathematics 27
If we rewrite the expression so that there is no radical in the denominator, it is called rationalizing the denominator.
Rationalizing the denominator is the process of eliminating the radical in the denominator.
Rationalizing the Denominator
Martin-Gay, Developmental Mathematics 28
Rationalize the denominator.
2
3
2
2
3 9
6
3
3
3
3
22
23
2
6
33
3
39
3 6
3
3
27
3 6
3
3 6 33 3 2
Rationalizing the Denominator
Example
Martin-Gay, Developmental Mathematics 29
Many rational quotients have a sum or difference of terms in a denominator, rather than a single radical.
•need to multiply by the conjugate of the denominator
•The conjugate uses the same terms, but the opposite operation (+ or ).
Conjugates
32
23
15
23
Martin-Gay, Developmental Mathematics 30
Martin-Gay, Developmental Mathematics 31
Rationalize the denominator.
32
23
332322
3222323
32
32
32
322236
1
322236
322236
Rationalizing the Denominator
Example