Transcript Square Root

Chapter 15
Roots and Radicals
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
15.1
Introduction to Radicals
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Radicands
The symbol
is called a radical or radical sign.
The expression within or under a radical sign is the
radicand.
A radical expression is an expression containing a
radical sign.
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Square Root
If a is a positive number, then
a is the positive square root of a and
 a is the negative square root of a.
Also, 0 = 0.
Note: A square root of a negative number is
not a real number.
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Square Roots
The reverse operation of squaring a number is taking
the square root of a number.
A number b is a square root of a number a if b2 = a.
In order to find a square root of a, you need a
number that, when squared, equals a.
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Perfect Squares
Square roots of perfect square radicands simplify to
rational numbers (numbers that can be written as a
quotient of integers).
Square roots of numbers that are not perfect squares
(like 7, 10, etc.) are irrational numbers. They cannot be
written as a quotient of integers.
If needed, you can find a decimal approximation for
these irrational numbers on a calculator. Otherwise,
leave them in radical form.
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Example
Find each square root.
a.
49 
b.
25 5

16 4
7
c.  4   2
d. Approximate
5
5 ≈ 2.236067977 to three decimal places.
5 ≈ 2.236
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Cube Root
The cube root of a real number a
3
a  b only if b 3  a
Note: a is not restricted to non-negative
numbers for cubes.
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Example
Find each cube root.
a.
3
27  3
b. 3 125  5
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Finding nth Roots
Other roots can be found, as well.
The nth root of a is defined as
n
a  b only if b n  a
If the index, n, is even, the root is NOT a real
number when a is negative.
If the index is odd, the root will be a real number.
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Example
Find each root.
a.
6
64  2
b.
5
32  2
c.
4
16
is not a real number since the index, 4, is
even and the radicand, –16, is negative.
There is no real number that when raised
to the 4th power gives –16.
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Simplifying Radicals Containing
Variables
Radicands might also contain variables and powers
of variables.
Now, if x is a negative number, like x = –2, then
x2
2
= (2) =
4 = 2, not –2, our original x.
To make sure that x2 simplifies to a nonnegative
number, we have the following.
For any real number a,
a2 = |a|
2
For example, (9) = |–9| = 9.
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Simplifying Radicals Containing
Variables
To avoid this confusion, for the rest of the chapter
we assume that if a variable appears in the
radicand of a radical expression, it represents
positive numbers only.
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Example
Find each root.
a. 64x10  8x 5
b.
c.
3
 8x 6   2x 2
2 20
25a b
 5ab10
3
4a
64
a
3
d. 
  3
9
b
b
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