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= + ÷ ± ∆ ∞ × π Σ f(x) % ≤ ! ≠ * |
RADICALS Square roots, cube roots and more!
When working with radicals, remember that these are numbers. They are not counting
numbers like 1,2,3, … but they are numbers. The difficulty is that they are usually
decimals that do not repeat and do not terminate. These irrational numbers can not be
written in decimal form. That is why the radical symbol 
was created, so that we could write them in an exact
form.
Note: We have used other symbols for the same reason. For example, the Greek letter pi written π is used to show the ratio of the circumference of a circle to the circle's diameter.
Along with writing them, we are concerned with the basic arithmetic operations add, subtract, multiply, and divide for radicals.
Add and Subtract two radicals
First don't add or subtract radicals to or from each other
Second, we have to know what are like terms and then addition and subtraction becomes easy, just combine like terms.
Ex. 
because 2 and 3 are like terms
Ex. 
is in simplified form because 
and 
are not like
terms and cannot be combined (added or subtracted)
Ex. 
is already simplified because they can not be
added (or subtracted)
If the radicals are the same then they can be added by multiplication.
and 
Multiply and Divide two radicals of the same degree
This is easy. 
and 
Ex. 
Ex. 
We can also simplify radicals themselves using this rule. When a number under the square root sign has a perfect square as a factor, then we can factor it out.
Ex. 
Ex. 
Ex. 
When multiplying square roots, sometimes we create perfect square factors.
For instance, 
and 
have no perfect square factors.
But their product is 
and 180 has three perfect square factors: 4,
9, and 36. This means that we can simply to get 
. This is a
good example of a radical problem that can be solved with different steps. For
instance, if you had noticed that 4 is a factor of 180 you may have simplified with the
following steps: 
. You got to the correct answer, just took a few extra steps. If you had
noticed that 9 was a perfect factor of 180 you might have followed these steps: 
.
We use radicals when finding some lengths with the Pythagorean Theorem.
Example: Given a square of side 5, what is the length of the diagonal?
The diagonal length
is found by using
. In the case of the square, a and b are the same so
Using the Pythagorean Theorem we find that the diagonal length is
and we showed above that
is simplified
.
We use this skill when working with the quadratic formula.
Suppose we have the function 
. Then 
So
how to have