Algebra Period 3 (Tuesday)

PRODUCT OF RADICALS 11-4
Basically, a radical is similar to variable in that you can multiply them,
but only add or subtract them if they are the exact same radicand (like terms)
So the product of the square root of 2 and the square root of 14 is the square root of 28

The square root of 28 can be simplified to 2 square root 7, read 2 "rad" 7

HELPFUL HINT:
If you are multiplying SQRT 250 times SQRT 50
I would suggest that you don't multiply 250 x 50 too quickly! Instead, factor 250 and factor 50
Then use the circling pairs method
This will actually save time generally (if you are not allowed to use a calculator!) because you won't end up with a humongous number that you will have to then simplify!
The way you would simplify is then to factor this big number!!!

So why not factor each factor first?!!!
Using my example above:
SQRT 250 times SQRT 50
SQRT(2 x 5 x 5 x 5) times SQRT(2 x 5 x 5) = SQRT(2 x 2 x 5 x 5 x 5 x 5 x 5)

(2 x 5 x 5) SQRT 5 = 50 SQRT 5

DIVIDING RADICALS 11-5

Just as you can multiply radicals, you can also divide them by either
1) separating the numerator from the denominator, or
2) simplifying the entire fraction underneath the radical.

HOW DO YOU KNOW WHICH METHOD TO USE:
Try both and see which one works best! (Examples below)

EXAMPLE OF TAKING THE QUOTIENT UNDER THE RADICAL APART:

Take apart fractions where either the numerator, the denominator, or both are perfect squares!

SQRT (3/16)= SQRT (3) = SQRT (3)

thiehtiehtiehtSQRT (16) thithii 4

SQRT (25/36) = SQRT (25) = 5

thisisthishtithithSQRT (36) th6

EXAMPLE OF SIMPLIFYING THE FRACTION UNDER THE RADICAL FIRST:
Sometimes, the fraction under the radical will simplify. If this is true, always do that first!

EXAMPLE: SQRT (27/3)

SQRT (27/3) = SQRT (9) = 3

If you took this fraction apart first and then tried to find the square root of each part, it's much more complicated:

SQRT (27/3) = SQRT (27) = SQRT (3 x 3 x 3) = 3 SQRT 3 = 3

thiththiththithtiSQRT 3 thithiSQRT 3thithithithiSQRT 3

You get the same answer, but with lots more steps!!!

AGAIN, SO HOW DO YOU KNOW WHICH TO DO????
Check both ways and see which works best!!!!!

RATIONALIZING THE DENOMINATOR

THE RULE: Simplified form has NO RADICALS IN THE DENOMINATOR.
(and you cannot change this rule even if you don't like it or think it makes sense!!!)
If you end up with a radical there, you must get rid of it by squaring whatever is under the radical.
Squaring it will result in the denominator becoming whatever was under the radical sign.
But you cannot do something to the denominator without doing the same thing to the numerator
(golden rule of fractions), so you must multiply the numerator by whatever you multiplied the denominator by.

RATIONALIZING THE DENOMINATOR EXAMPLE:
SQRT 7
SQRT 3

There is nothing you can simplify, whether you put it together or take it apart!
But you can't leave it this way because the rule is that you can't leave the
SQRT(3) in the denominator.
You need to multiply both numerator and denominator by SQRT (3) to get it out of there:

SQRT 7 = SQRT 7 * SQRT 3 = SQRT (7 * 3) = SQRT 21
SQRT 3 thiSQRT 3 thiSQRT 3 thithithi3thithithithi3

Note that you cannot cross cancel the 3 in denominator with 21 in numerator
because one is a square root and the other is not (they are unlike terms!)
For example, if you were to simplify SQRT(9)/3, you would get 3/3 = 1

If you had simplified the 3 with the 9 first, you would have incorrectly gotten SQRT (3)