Algebra 1
Section 11.1
Square Roots
Squaring a number and taking
the square root of the result
are similar to inverse
operations, such as addition
and subtraction.
The square roots of 9 are 3
and -3.
Definition
A square root is one of a
number’s two equal factors.
The symbol for a square root is
the radical sign,
.
Definition
The expression under the
radical sign is called the
radicand.
Square Roots
The square root of 9 has two
values: ±3.
To avoid confusion, we use the
radical sign to indicate the
positive square root, called the
principal root.
Square Roots
If the negative square root is
desired, a negative sign is
placed in front of the radical
sign.
9 =3
-
9 = -3
Example 1
4 =2
- 16 = -4
Roots
To indicate roots other than
square roots, an index is added
to the radical sign.
Index
n
Definitions
3
A cube root,
, is one of a
number’s three equal factors.
n
An nth root,
, is one of a
number’s n equal factors.
If
23
= 8, then
3
8 = 2.
Example 2
3
4
125 =
3
53 = 5
81 =
4
34 = 3
Roots
The even root of a negative
number is not a real number
since any number to an even
power is positive.
The odd root of a negative
number is a negative real
number.
Example 3
a.
b.
c.
-4 is not a real number.
3 - 27
125
5
=
-32 =
3
5
3 3 =-3
-( 5 )
5
(-2)5 = -2
Rational and Irrational Numbers
While some radicals are
rational numbers, many
radicals are irrational numbers.
Irrational numbers are
nonrepeating, nonterminating
decimals.
Rational and Irrational Numbers
Irrational numbers can be
either stated exactly as radicals
or approximated by rounding to
a designated decimal place.
Approximation
Since 4 < 7 < 9,
4 <
2 <
7 <
9
7 < 3
7 ≈ 2.645751311
7 ≈ 2.65
Example 4
121 < 129 < 144
Therefore,
11 <
129 < 144
And on a calculator,
129 ≈ 11.36
Example 5
27 < 50 < 64
Therefore,
3<
3
50 < 4
And on a calculator,
3
50 ≈ 3.684
Roots and Radicals
The definitions of roots and
powers allow you to convert an
equation from exponential to
radical form and vice versa.
In general, bn = a is equivalent
n
to a = b.
Roots and Radicals
The definition of exponents is
not limited to integers.
You can convert between the
radical form and the
exponential form of any
expression by using the
following definition.
Definition
The exponential expression
is equivalent to the radical
b a
b
x or ( x )a .
b
a
x
Example 6
5
2
3
=
5 3
x
=
3
52
3
x5
3
= 25
Example 7
1
32
4
3
x4
=
2
34
7x3y2 =
3
x4
1
74
=
4
32x3
3 2
x4y4
=
=
1
74
4
9x3
3 1
x4y2
Homework:
pp. 435-436