**Properties**

for all non- negative numbers x

for all non-negative numbers x

However , if x happens to be negative, then squaring

it will produce a positive number, which will have a

positive square root , so

for all real numbers x

You don’t need the absolute value sign if you

already know that x is positive. For example,

, and saying anything about the absolute

value of 2 would be superfluous. You only need

the absolute value signs when you are taking the

square root of a square of a variable , which may

be positive or negative.

The square root of a negative number is

undefined, because anything times itself will

give a positive (or zero) result.

(your calculator will probably

say ERROR).

** Note:** Zero has only one square root (itself). Zero

is considered neither positive nor negative.

**WARNING:** Do not attempt to do something like

the distributive law with radicals :

**(WRONG) **or

** (WRONG)**.

This is a violation of the order of operations . The

radical operates on the result of everything inside of

it, not individual terms . Try it with numbers to see:

**(CORRECT)**

But if we (incorrectly) do the square roots first, we

get

**(WRONG)**

However, radicals do distribute over products :

and

provided that both a and b are non-negative

(otherwise you would have the square root of a

negative number).

**Perfect Squares**

Some numbers are perfect squares, that is, their

square roots are integers:

0, 1, 4, 9, 16, 25, 36, etc.

It turns out that all other whole numbers have

irrational square roots:

, etc. are all irrational numbers.

The square root of an integer is either perfect or

irrational

**C. Simplifying Radical Expressions**

for all real numbers

if both x and y are non-negative, and

if both x and y are non-negative, and y is
not

zero

**WARNING:** Never cancel something inside a

radical with something outside of it:

**
WRONG! **If you did this you would be

canceling a 3 with , and they are certainly not the

same number.

The general plan for reducing the radicand is to

remove any perfect powers. We are only considering

square roots here, so what we are looking for is any

factor that is a perfect square. In the following

examples we will assume that x is positive.

**Example:**

In this case the 16 was recognized as a perfect square

and removed from the radical, causing it to become

its square root, 4.

**Example:**

Although is not a perfect square, it has a factor of

, which is the square of x.

**Example:**

Here the perfect square factor is
, which is the

square of .

**Example:**

In this example we could take out a 4 and a factor of

, leaving behind a 2 and one factor of x.

The basic idea is to factor out anything that is

“square-rootable” and then go ahead and square

root it.

**D. Rationalizing the Denominator **

One of the “rules” for simplifying radicals is that you

should never leave a radical in the denominator of a

fraction. The reason for this rule is unclear (it

appears to be a holdover from the days of slide

rules), but it is nevertheless a rule that you will be

expected to know in future math classes. The way to

get rid of a square root is to multiply it by itself,

which of course will give you whatever it was the

square root of. To keep things legal, you must do to

the numerator whatever you do to the denominator,

and so we have the rule:

**If the Denominator is Just a Single Radical**

**Multiply the numerator and denominator by the**

denominator

**Example:**

Note: If you are dealing with an nth root instead

of a square root, then you need n factors of that

root in order to make it go away. For instance, if

it is a cube root (n = 3), then you need to

multiply by two more factors of that root to give

a total of three factors.

**If the Denominator Contains Two Terms**

If the denominator contains a square root plus some

other terms, a special trick does the job. It makes use

of the difference of two squares formula:

(a + b)(a – b) = a^{2} – b^{2}

Suppose that your denominator looked like a + b,

where b was a square root and a represents all the

other terms. If you multiply it by a – b, then you will

end up with the square of your square root, which

means no more square roots. It is called the

conjugate when you replace the plus with a minus (or

vice-versa). An example would help.

**Example:**

Given:

Multiply numerator and denominator by the

conjugate of the denominator:

Multiply out: