1. 7.1 – Radicals
Radical Expressions
Finding a root of a number is the inverse operation of raising
a number to a power.
This symbol is the radical or the radical sign
n
a
index radical sign
radicand
The expression under the radical sign is the radicand.
The index defines the root to be taken.
2. Radical Expressions
The symbol represents the negative root of a number.
The above symbol represents the positive or principal
root of a number.
7.1 – Radicals
3. Square Roots
If a is a positive number, then
a is the positive square root of a and
100
a
is the negative square root of a.
A square root of any positive number has two roots – one is
positive and the other is negative.
Examples:
10
25
49
5
7
0.81 0.9
36
6
9
non-real #
8
x 4
x
7.1 – Radicals
4. Rdicals
Cube Roots
3
27
A cube root of any positive number is positive.
Examples:
3
5
4
3
125
64
3
8
2
A cube root of any negative number is negative.
3
a
3 3
x x
3 12
x
4
x
7.1 – Radicals
5. nth
Roots
An nth
root of any number a is a number whose nth
power is a.
Examples:
2
4
81 3
4
16
5
32
2
4
3 81
4
2 16
5
2
32
7.1 – Radicals
6. nth
Roots
4
16
An nth
root of any number a is a number whose nth
power is a.
Examples:
1
5
1
Non-real number
6
1
Non-real number
3
27
3
7.1 – Radicals
7. 7.2 – Rational Exponents
The value of the numerator represents the power of the
radicand.
Examples:
:
n
m
a
of
Definition
The value of the denominator represents the index or root of
the expression.
n m
a or m
n
a
3
1
27
25
2
1
25 3
5 3
27
7
2
1
2
x
3
4
2
3
4 64
7 2
1
2
x
8
8. 7.2 – Rational Exponents
More Examples:
:
n
m
a
of
Definition n m
a or m
n
a
3
2
3
2
27
1
3
2
27
1
3 2
3 2
27
1
9
1
3
3
729
1
3
2
3
2
27
1
3
2
27
1
2
3
2
3
27
1
9
1
2
2
3
1
or
9. 7.2 – Rational Exponents
Examples:
:
n
m
a
of
Definition
n m
a
1
m
n
a
1
2
1
25
1
2
1
25
25
1
5
1
3
2
1
x
3
2
x 3 2
1
x 2
3
1
x
n
m
a
1
or or
or
10. 7.2 – Rational Exponents
Use the properties of exponents to simplify each expression
3
5
3
4
x
x 3
9
x
3
x
10
1
5
3
x
10
1
5
3
x
x
10
1
10
6
x 10
5
x
4
2
3x
4 2
81x 2
1
3x
3
5
3
4
x
2
1
x
3 2
12
x
x 12
8
12
1
x 12
9
x 4
3
x
3
2
12
1
x
x
11. 40
Examples:
4 10
If and are real numbers, then a b
a b a b
Product Rule for Square Roots
2 10
7 75 7 25 3
7 5 3
35 3
7.3 – Simplifying Rational Expressions
17
16x
x
x16
16 x
x8
4
3 17
16x
3 2
15
2
8 x
x 3 2
5
2
2 x
x
10
4
3
25
7
12. 16
81
Examples:
2
5
4
9
45
49
a
If and are real numbers and 0,then
b
a
a b b
b
Quotient Rule for Square Roots
2
25
9 5
7
3 5
7
16
81
2
25
45
49
7.3 – Simplifying Rational Expressions
13. 15
3
90
2
a
If and are real numbers and 0,then
b
a
a b b
b
3 5
3
3 5
3
5
9 10
2
9 2 5
2
9 2 5
2
3 5
7.3 – Simplifying Rational Expressions
19. 4 3 3
9 36
x x x
Simplifying Radicals Prior to Adding or Subtracting
6 6
3 3
10 81 24
p p
2 2 2
3 6
x x x x x
2
3 6
x x x x x
2
3 5
x x x
6 6
3 3
10 27 3 8 3
p p
2 2
3 3
10 3 3 2 3
p p
2 3
28 3
p
2 2
3 3
30 3 2 3
p p
7.4 – Adding, Subtracting, Multiplying Radical
Expressions
20. 5 2
7 7
10 2
x x
If and are real numbers, then a b
a b a b
10
49 7
6 3
18 9 2
3 2
2
20x 2
4 5x
2 5
x
7.4 – Adding, Subtracting, Multiplying Radical
Expressions
21.
7 7 3
7 7 7 3
49 21
5 3 5
x x
5 3
x x
7 21
2
5 3 25
x x
5 3 5
x x
5 15
x x
2
3 5 15
x x x
2
3 5 15
x x x
7.4 – Adding, Subtracting, Multiplying Radical
Expressions
22.
3 6 3 6
2
5 4
x
9 6 3 6 3 36
3 36
33
5 4 5 4
x x
2
25 4 5 4 5 16
x x x
5 8 5 16
x x
7.4 – Adding, Subtracting, Multiplying Radical
Expressions
