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 s n a inde x radical sign radica nd The expression under the radical sign is the radicand. The index defines the root to be taken.
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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.
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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