Download b a 6 36, = - - = 6 4 4 4 4 - 9 16 6

Section 10.1: Radical Expressions And Graphs
§1 Find Square Roots
By definition, a number b is called a square root of a if b2  a . We use the radical symbol √ to denote the
square root of a number. Hence, if 62  36, then the square root of 36 is 6. Note also that (6)(6)  36 , so 6 is a square root of 36. Hence a positive number always has two square roots – the positive or principal square
root, and the negative square root.
It’s very important to note, however, that the radical symbol √ always represents the positive square root.
Hence we never say that
36  6 ; we say that 36  6 only.
The number inside the radical symbol is called the radicand and the entire expression is called the radical.
If we square a radical, the result is the radicand itself. Hence something like

with a negative expression. Something like  4

2
 4
2
 4 . You must be careful
 4 , not -4.
PRACTICE
1) Find
25 and  144 and
2) Find the square of
9
16
3 and the square of  6
§2 Irrational Numbers And Perfect Squares
Any number whose square root is rational is called a perfect square. It’s a very good idea to memorize the first
20 perfect squares. For example, since 152  225, then
225  15 . So 225 is called a perfect square.
A number that is not rational is called irrational. Many square roots are irrational, like
number is neither rational or irrational, then it is not a real number, like
3 and 10 . If a
4 .
§3 Finding Higher Roots
Remember, if
a  b, then b2  a . Note that the power of b is 2. That’s why the radical symbol
is called
the square root. We see that finding the square root of a number is the opposite (or inverse) of squaring a
number. But what about if we cube a number? Take something like 43  64 . What would be its inverse? Note
that here the power is 3. Hence the inverse is not a square root, we call it the cube root. Hence if 43  64,
then 3 64  4 . More generally, if a n  b, then n b  a . In the radical form, the number n is called the index or
order of the radical. This is a term you definitely want to know.
PRACTICE
3
8
3) Find
3
8 and
4) Find
4
16 and 4 16
§4 Finding nth Roots Of nth Powers
The expression
a 2 gets to be a little tricky. For example, what is the difference between
Nothing, actually. They both equal 6. However, a lot of people think that
reality, we say that
62 and (6) 2 ?
(6)2  6 . This is not the case. In
a 2  a . This is telling us that in this case, a may be a negative number.
For higher indexes, the same property applies but with a slight difference if the index is even or odd. If n is a
positive even integer, then
n
a n  a , while if n is a positive odd integer,
n
an  a .
PRACTICE
5) Simplify
4
(5)4 and
6) Simplify
4
28
3
(5)3
NOTE: We will not go over the section on graphing or using a calculator to find roots.