7.1 – radicals radical expressions finding a root of a number is the inverse operation of raising...
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7.1 – RadicalsRadical 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.
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
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
70.81 0.9
36 6
9 non-real #
8x 4x
7.1 – Radicals
RdicalsCube Roots
3 27
A cube root of any positive number is positive.
Examples:
35
43
125
643 8 2
A cube root of any negative number is negative.
3 a
3 3x x 3 12x 4x
7.1 – Radicals
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
43 81
42 16
52 32
7.1 – Radicals
nth Roots
4 16
An nth root of any number a is a number whose nth power is a.
Examples:
15 1
Non-real number
6 1 Non-real number
3 27 3
7.1 – Radicals
7.2 – Rational Exponents
The value of the numerator represents the power of the radicand.
Examples:
:nm
aofDefinition
The value of the denominator represents the index or root of the expression.
n ma or mn a
31
272521
25 35 3 27
72
12 x
3423
4 64
7 212 x
8
7.2 – Rational Exponents
More Examples:
:nm
aofDefinition n ma or mn a
32
32
27
132
27
1
3 2
3 2
27
19
13
3
729
1
32
32
27
132
27
1
23
23
27
1
9
1 2
2
3
1
or
7.2 – Rational Exponents
Examples:
:nm
aofDefinition
n ma
1
mn a
1
21
25
12
125
25
1
5
1
32
1
x3
2x 3 2
1
x 23
1
x
nma
1or or
or
7.2 – Rational ExponentsUse the properties of exponents to simplify each expression
35
34xx 3
9x
3x
101
53 x
101
53
x
x10
110
6 x 10
5x
42
3x4 281x 21
3x
35
34 x
21x
3 212 xx 128
121 x 12
9x 4
3x3
212
1xx
40
Examples:
4 10
If and are real numbers, then a ba 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
1716x xx1616 xx84
3 1716x 3 21528 xx 3 25 22 xx
104
3257
16
81
Examples:
2
5
4
9
45
49
aIf and are real numbers and 0, then
b
aa 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
15
3
90
2
aIf and are real numbers and 0, then
b
aa 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
11x
Examples:
77
25
y
8
27
x
67
25
y y
3 7
5
y y
10x x 5x x
418x 49 2x 23 2x
8
9 3
x
4
3 3
x8
27
x
7.3 – Simplifying Rational Expressions
3 88
Examples:
381
8
310
27
3
3
81
8
3 27 3
2
3 8 11 32 11
3 10
3
3
3
10
27
33 3
2
7.3 – Simplifying Rational Expressions
3 3 727m n 3 3 63 m n n 2 33mn n
One Big Final Example
12 4 185 64x y z
10 2 4 15 35 32 2x x y z z
2 3 2 4 352 2x z x y z
7.3 – Simplifying Rational Expressions
5 3x x
Review and Examples:
6 11 9 11
8x
15 11
12 7y y 5y
7 3 7 2 7
7.4 – Adding, Subtracting, Multiplying Radical Expressions
27 75
Simplifying Radicals Prior to Adding or Subtracting
3 20 7 45
9 3 25 3
3 4 5 7 9 5
3 3 5 3 8 3
3 2 5 7 3 5
6 5 21 5 15 5
36 48 4 3 9 6 16 3 4 3 3
6 4 3 4 3 3 3 8 3
7.4 – Adding, Subtracting, Multiplying Radical Expressions
4 3 39 36x x x
Simplifying Radicals Prior to Adding or Subtracting
6 63 310 81 24p p
2 2 23 6x x x x x
23 6x x x x x 23 5x x x
6 63 310 27 3 8 3p p
2 23 310 3 3 2 3p p 2 328 3p
2 23 330 3 2 3p p
7.4 – Adding, Subtracting, Multiplying Radical Expressions
5 2
7 7
10 2x x
If and are real numbers, then a ba b a b
10
49 7
6 3 18 9 2 3 2
220x 24 5x 2 5x
7.4 – Adding, Subtracting, Multiplying Radical Expressions
7 7 3 7 7 7 3 49 21
5 3 5x x
5 3x x
7 21
25 3 25x x 5 3 5x x
5 15x x
2 3 5 15x x x
2 3 5 15x x x
7.4 – Adding, Subtracting, Multiplying Radical Expressions
3 6 3 6
2
5 4x
9 6 3 6 3 36 3 36 33
5 4 5 4x x
225 4 5 4 5 16x x x
5 8 5 16x x
7.4 – Adding, Subtracting, Multiplying Radical Expressions