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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
radical sign
index
n
a
radicand
The expression under the radical sign is the radicand.
The index defines the root to be taken.
7.1 – Radicals
Radical Expressions
The above symbol represents the positive or principal
root of a number.
пЂ­
The symbol represents the negative root of a number.
7.1 – Radicals
Square Roots
A square root of any positive number has two roots – one is
positive and the other is negative.
If a is a positive number, then
a
пЂ­ a
is the positive square root of a and
is the negative square root of a.
Examples:
100 пЂЅ 1 0
25
49
пЂЅ
5
7
пЂ­ 36 пЂЅ пЂ­ 6
0.81 пЂЅ 0 .9
x пЂЅ x
4
пЂ­9 пЂЅ
non-real #
8
7.1 Rdicals
– Radicals
Cube Roots
3
a
A cube root of any positive number is positive.
A cube root of any negative number is negative.
Examples:
3
3
27 пЂЅ 3
x пЂЅ
3
x
3
3
пЂ­8 пЂЅ пЂ­2
x
12
пЂЅ x
4
3
125
64
пЂЅ
5
4
7.1 – Radicals
nth Roots
An nth root of any number a is a number whose nth power is a.
Examples:
3 пЂЅ 81
4
81 пЂЅ 3
2 пЂЅ 16
4
16 пЂЅ 2
5
пЂ­ 32 пЂЅ пЂ­ 2
4
4
пЂЁ пЂ­2 пЂ©
5
пЂЅ пЂ­32
7.1 – Radicals
nth Roots
An nth root of any number a is a number whose nth power is a.
Examples:
3
5
пЂ­1 пЂЅ пЂ­1
4
пЂ­ 16 пЂЅ Non-real number
6
пЂ­1 пЂЅ
Non-real number
пЂ­ 27 пЂЅ пЂ­ 3
7.2 – Rational Exponents
Definition
of a
m
n
n
:
a
m
or
пЂЁ aпЂ©
m
n
The value of the numerator represents the power of the
radicand.
The value of the denominator represents the index or root of
the expression.
Examples:
1
1
25
3
4
4
2
пЂЁ 2 x пЂ« 1пЂ©
2
7
27
5
25
2
3
8
64
7
пЂЁ 2 x пЂ« 1пЂ©
3
2
3
27
3
7.2 – Rational Exponents
Definition
of a
m
n
n
:
a
m
пЂЁ aпЂ©
or
n
m
More Examples:
пѓ¦ 1 пѓ¶
пѓ§
пѓ·
пѓЁ 27 пѓё
2
2
3
3
3
1
2
3
3
27
1
2
27
3
3
2
1
1
729
9
or
пѓ¦ 1 пѓ¶
пѓ§
пѓ·
пѓЁ 27 пѓё
2
2
3
1
3
2
27
пЂЁ1пЂ©
3
3
пЂЁ
3
27
2
пЂ©
пЂЁ1 пЂ©
2
пЂЁ3 пЂ©
2
2
1
9
7.2 – Rational Exponents
Definition
пЂ­
of a
m
n
1
:
or
m
a
1
n
n
a
or
m
25
x
пЂ­2
1
2
1
25
2
1
3
2
x
3
3
1
1
25
5
1
1
x
2
or
пЂЁ xпЂ©
3
пЂЁ aпЂ©
n
Examples:
пЂ­1
1
2
m
7.2 – Rational Exponents
Use the properties of exponents to simplify each expression
4
3
x
пѓ—x
5
3
x
9
4 пЂ«5
3
3
x
x
3
3
3
5
x
x
1
x
4
12
10
81 x
2
xпѓ—
3
6
3 пЂ­1
5
10
x
10
x
2
5
10
x
1
10
x
2
1
2
3x
пЂ­1
3x
4
2
1
x
2
12
x
3
1
x
12
пЂ«8
9
12
x
3
12
x
4
7.3 – Simplifying Rational Expressions
Product Rule for Square Roots
If
a and
b are real num bers, then
a пѓ—b пЂЅ
aпѓ— b
Examples:
40 пЂЅ
4 пѓ— 10 пЂЅ
4 10 пЂЅ 2 10
7 75 пЂЅ 7 25 пѓ— 3 пЂЅ 7 25
3
16 x
17
пЂЅ
16 x
17
пЂЅ
16 x x пЂЅ 4 x
16
3
8пѓ— 2x x
15
2
8
пЂЅ 2x
3 пЂЅ 7 пѓ—5 3 пЂЅ
x
5 3
2x
2
35 3
7.3 – Simplifying Rational Expressions
Quotient Rule for Square Roots
If
a
b are real num bers and b п‚№ 0, then
a and
b
Examples:
16
пЂЅ
81
45
49
16
пЂЅ
81
пЂЅ
45
49
пЂЅ
4
2
9
25
9 пѓ—5
7
пЂЅ
3 5
7
пЂЅ
2
25
пЂЅ
2
5
пЂЅ
a
b
7.3 – Simplifying Rational Expressions
If
a and
b are real num bers and b п‚№ 0, then
a
b
15
пЂЅ
3пѓ—5
3
3
90
9 пѓ— 10
2
пЂЅ
2
пЂЅ
3пѓ— 5
пЂЅ
5
3
пЂЅ
9 пѓ—2 пѓ—5
2
пЂЅ
9пѓ— 2пѓ— 5
2
пЂЅ 3 5
пЂЅ
a
b
7.3 – Simplifying Rational Expressions
Examples:
x
11
18 x
27
x
8
7y
7
25
пЂЅ
4
пЂЅ
x
пЂЅ
пѓ—x пЂЅ
10
9 пѓ—2x
27
x
4
x
пЂЅ
8
x
6
пЂЅ
3x
9 пѓ—3
пЂЅ
7пѓ—y y
25
5
пЂЅ
x
2
2
пЂЅ
3 3
8
y
x
3
7y
5
4
7.3 – Simplifying Rational Expressions
Examples:
3
3
88 пЂЅ
3
10
3
пЂЅ
3
27
3
3
27 m n
3
7
81
3
8
пЂЅ
3
8 пѓ— 11 пЂЅ
3
10
пЂЅ
81
8
10
3
27
пЂЅ
2 3 11
3 m n n пЂЅ 3mn
3
3
3
пЂЅ
6
27 пѓ— 3
2
пЂЅ
2 3
33 3
2
n
7.3 – Simplifying Rational Expressions
One Big Final Example
5
5
12
4
64 x y z
18
пЂЅ
32 пѓ— 2 x x y z z пЂЅ
10
2
2x z
3 5
2
4
2
15
4
2x y z
3
3
7.4 – Adding, Subtracting, Multiplying Radical
Expressions
Review and Examples:
5x пЂ« 3x пЂЅ 8x
12 y пЂ­ 7 y пЂЅ 5 y
6 11 пЂ« 9 11 пЂЅ 15 11
7 пЂ­ 3 7 пЂЅ пЂ­2 7
7.4 – Adding, Subtracting, Multiplying Radical
Expressions
Simplifying Radicals Prior to Adding or Subtracting
27 пЂ«
75 пЂЅ
9 пѓ—3 пЂ«
25 пѓ— 3 пЂЅ 3 3 пЂ« 5 3 пЂЅ 8 3
3 20 пЂ­ 7 45 пЂЅ 3 4 пѓ— 5 пЂ­ 7 9 пѓ— 5 пЂЅ 3 пѓ— 2 5 пЂ­ 7 пѓ— 3 5 пЂЅ
6 5 пЂ­ 21 5 пЂЅ
36 пЂ­
48 пЂ­ 4 3 пЂ­
9 пЂЅ
пЂ­ 15 5
6 пЂ­ 16 пѓ— 3 пЂ­ 4 3 пЂ­ 3 пЂЅ
6пЂ­4 3 пЂ­4 3 пЂ­3 пЂЅ 3пЂ­8 3
7.4 – Adding, Subtracting, Multiplying Radical
Expressions
Simplifying Radicals Prior to Adding or Subtracting
9x пЂ­
4
36 x пЂ«
3
x пЂЅ
3x пЂ­ 6 x x пЂ«
3
2
x x пЂЅ
2
2
3x пЂ­ 6 x x пЂ« x x пЂЅ 3x пЂ­ 5x x
2
2
10 3 81 p пЂ­
6
10 пѓ— 3 p
2 3
3
24 p пЂЅ
3пЂ­2p
6
2 3
10 3 27 пѓ— 3 p пЂ­
6
3 пЂЅ
28 p
30 p
2 3
3
2 3
3
3пЂ­2p
8 пѓ—3 p пЂЅ
6
2 3
3 пЂЅ
7.4 – Adding, Subtracting, Multiplying Radical
Expressions
If
a and
b are real num bers, then
7пѓ— 7 пЂЅ
49 пЂЅ 7
5пѓ— 2 пЂЅ
10
6пѓ— 3 пЂЅ
18 пЂЅ 9 пѓ— 2 пЂЅ
10 x пѓ— 2 x пЂЅ
20 x пЂЅ
2
a пѓ—b пЂЅ
3 2
4 пѓ—5x пЂЅ
2
2x 5
aпѓ— b
7.4 – Adding, Subtracting, Multiplying Radical
Expressions
7
пЂЁ
пЂ©
7пЂ­
7пѓ— 7пЂ­
3 пЂЅ
7пЂ­
5x
пЂЁ
пЂ©
x пЂ­3 5 пЂЅ
49 пЂ­
7пѓ— 3пЂЅ
21 пЂЅ
21
5 x пЂ­ 3 25 x пЂЅ
2
x 5 пЂ­ 3пѓ—5 x пЂЅ
x 5 пЂ­ 15 x
пЂЁ
xпЂ«
5
пЂ©пЂЁ
xпЂ­
пЂ©
3 пЂЅ
x пЂ­
2
x пЂ­
2
3x пЂ«
3x пЂ«
5 x пЂ­ 15
5 x пЂ­ 15 пЂЅ
7.4 – Adding, Subtracting, Multiplying Radical
Expressions
пЂЁ
3пЂ«6
пЂ©пЂЁ
пЂ©
3пЂ­6 пЂЅ
9 пЂ­ 6 3 пЂ« 6 3 пЂ­ 36 пЂЅ 3 пЂ­ 3 6 пЂЅ
пЂ­33
пЂЁ
5x пЂ« 4
пЂ©
2
пЂЅ
пЂЁ
5x пЂ« 4
пЂ©пЂЁ
пЂ©
5x пЂ« 4 пЂЅ
25 x пЂ« 4 5 x пЂ« 4 5 x пЂ« 16 пЂЅ
2
5 x пЂ« 8 5 x пЂ« 16
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