Algebra Period 3

SIMPLIFYING RADICALS 11-3

SIMPLIFYING NONPERFECT NUMBERS UNDER THE RADICAL:
A simplified radical expression is one where there is no perfect square left under the radical sign

You can factor the expression under the radical to find any perfect squares in the number:
EXAMPLE: √50 = √(25 * 2)
Next, simplify the sqrt of the perfect square and leave the nonperfect factor under the radical:
√(25 * 2) = √25 * √2 = 5√2
We usually read the answer as "5 rad 2"

HELPFUL HINTS:
When you are factoring the radicand,
you're looking for the LARGEST PERFECT SQUARE
that is a FACTOR of the radicand.
So start with:
Does 4 go into it?
Does 9 go into it?
Does 16 go into it?
Does 25 go into it?
etc.

Another method: Inverted Division or Factor Trees
Factor the radicand completely into its prime factors (remember this from Pre-Algebra?)
Find the prime factorization either way in order from least to greatest.
Circle factors in PAIRS
Every time you have a pair, you have a factor that is squared!
Then, you can take that factor out of the radical sign.
Remember that you are just taking one of those factors out!

Example: √ 250
Prime factorization = 2 x 5 x 5 x 5
Circle the first two 5's
5 x 5 is 25 and so you can take the square root of 25 = 5 out of the radicand
Everything else is not in a pair (squared) so it must remain under the radical

Final answer: √ 250 = 5√(2 x 5) = 5 √10

VARIABLES UNDER THE SQUARE ROOT SIGN:
An even power of a variable just needs to be divided by two to find its square root
EXAMPLE: √x10 = x5
We saw this already in factoring!!!

EXAMPLE: √75x10
√ [(25)(3)(x10 )]
or
√[(5)(5)](3)(x5x5)
Simplified, you can pull out a factor of 5 and x5
5x5√3

If the variable has an odd power:
If you have an odd power variable, simply express it as the
(even power one below)(variable to the 1 power)

EXAMPLE: √x5
5 is an odd power, so go down to the next even power (4)
√ (x4 x)
Now you can find the square root of the even power and the 1 power is just left under the √
x2√x

YOU SHOULD NEVER HAVE A VARIABLE OF MORE THAN THE 1 POWER
UNDER THE SQUARE ROOT SIGN!
IF YOU DO,
YOU HAVE NOT SIMPLIFIED ALL THE WAY!

FACTORING A GCF FIRST,
THEN FINDING A BINOMIAL SQUARED:
Sometimes you will need to factor what's under the radical before you start to simplify
Example: √(3x2 + 12x + 12)
First factor out a 3:
√[3 (x2 + 4x + 4) ]
Now factor the trinomial:
√ [3 (x + 2) (x + 2) ]
The (x + 2)2 is a perfect square so
√ [3(x + 2)2] = (x + 2) √ 3