| Surviving College Algebra |
| "When all you want is the grade" |
 | |  | |  | |  |
This 
is called a
radical, it means “square root of”.
A
problem like 
, would be read as “the square root of n”.
Furthermore,
the square root is trying to find out what number times itself is equal to n.
Example
Look
for the number that when multiplied by itself, will be 9. The easiest method is to use a
calculator. By using a calculator, find
out that the correct answer is 3, because 3(3)=9. Now understand that -3 could be a solution
also. Since (-3)(-3
) is equal to 9 also. So the correct
answer is ± 3.
±3
Answer!
Note-
Read as
“Plus or minus 3”, which means a positive or negative 3.
Example
Ask
the question, “What number times itself will be 36?”.
± 6
Answer!
Example
Ask
the question, “What number times itself will be 169?”.
±
13
Answer!
In
the previous examples, the numbers that were underneath the radicals are called
“perfect squares”. In other words, their
square roots are integers. Here is a
table of important radicals and numbers that need to be recognized.
= 0
=
1
=
2
=
3
= 4
=
5
=
6
=
7
= 8
=
9
=
10
=
11
=
12
The
list goes on and on forever but the numbers to be familiar with are 0,1, 4,9,16,25, 36, 49, 64, 81, 100, 121, and 144. These numbers are called perfect squares
because the square root of each one is an integer.
Multiplication rule of Radicals
= 
Or
= 
When
simplifying radicals, use perfect
squares or leave them how they are.
Those are the only two choices.
Example
This
number is not a perfect square but it can be written as the product of a
perfect square and some other number. 12 is the same thing as (4)(3). Therefore write
it as
.
By
the multiplication rule of radicals, the following is true.
= 
is = 2.
2
Answer!
At
this point see if the number under the radical, 3, can break down with a
perfect square (other than 1). Since it cannot, leave it like it is.
Example
24 is the same as 4 (6).
Notice that 4 is a perfect square, break down the original number (24)
using at least one perfect square.
=
2
2
Answer!
Note
– Always
check to see if the number underneath the radical will break down. In this case, it is 6. Since it will not break down with a perfect
square, leave it like it is.
Example
72
can be written as 36 (2).
=
6
6
Answer!
or
72
can be written as 9 (8).
=
3
3
Break
down 8 into 4 (2)
3
The
square root of 4 is 2
3
(2)
3(2)
= 6
6
Answer!
Note- It does not matter which
method is used on the previous example, they will both produce the same
answer. Make sure that the number
underneath the radical cannot be broken down with a perfect square. The larger the perfect square that is used
for the first step, the shorter the problem will be.
Example
800
can be written as 8 (100)
=
10
10
8
can be written as 4(2).
10
= 2, and 10(2) = 20.
20
Answer!
Always
remember that there are many ways to work the same problem but there is only
one correct answer!
Here
are a couple of easy rules to work with variables when they are underneath a
radical.
| Radicals |