Algebra Period 3 (Monday)

REAL NUMBERS 11-1
as opposed to IMAGINARY numbers! (Seriously!)
√ is the symbol for radicals. We use this symbol (without any small number on the radical) to represent square root (At times in the post I may need to use SQ RT to refer to square root)

Square rooting "undoes" squaring!
It's the inverse operation: Just as subtraction undoes addition and Just as division undoes multiplication

EXAMPLES:
If you square a square root:
(√243)² = 243 (what you started with)
If you square root something squared:
√243² = 243 (what you started with)

If you multiply a square root by the same square root:
(√243)(√243) = 243 (what you started with)

IN SUMMARY:
(√243)² = √2432 = (√243)(√243) = 243
1) RADICAL sign: The root sign, which looks like a check mark.
If there is no little number on the radical, you assume it's the square root
But many times there will be a number there and then you are finding the root that the number says.
For example, if there is a 3 in the "check mark," you are finding the cubed root.
One more example: The square root of 64 is 8. The cubed root of 64 is 3. The 6th root of 64 is 2.

2)RADICAND : Whatever is under the RADICAL sign
In the example above, 64 was the radicand in every case.

3) ROOT (the answer): the number/variable that was squared (cubed, raised to a power)
to get the RADICAND (whatever is under the radical sign) In the example above, the roots were 8, 3, and 2.

4) SQUARE ROOTS: (What we primarily cover in Algebra I) The number that is squared to get to the radicand.
Every POSITIVE number has 2 square roots - one positive and one negative.
Example: The square root of 25 means what number squared = 25
Answer: Either positive 5 squared OR negative 5 squared = 25

5) PRINCIPAL SQUARE ROOT: The positive square root.
Generally, the first section just asks for the principal square root unless there is a negative sign in front of the radical sign.
6) ± sign in front of the root denotes both the positive and negative roots at one time!
Example: √ 25 = ±5

7) ORDER OF OPERATIONS with RADICALS: Radicals function like parentheses when there is an operation under the radical. In other words, if there is addition under the radical, you must do that first (like you would do parentheses first) before finding the root.
EXAMPLE: √ (36 + 64) = 10 not 14!!!!
First add 36 + 64 = 100
Then find √100 = 10

Radicals by themselves function as exponents in order of operations
(that makes sense because they undo exponents).
Actually, roots are FRACTIONAL EXPONENTS!
Square roots = 1/2 power,
Cubed roots = 1/3 power,
Fourth roots = 1/4 power, etc.
So √25 = 25^1/2 = 5
EXAMPLE: 3 + 4√25
you would do powers first...in this case square root of 25 first!
3 + 4(5)
Now do the multiplication
3 + 20
Now do the addition
23

8) THE SQUARE ROOT OF ANYTHING SQUARED IS ITSELF!!!
EXAMPLE: √ 5² = 5
√ (a -7)² = a - 7
RATIONAL SQUARE ROOTS:
Square roots of perfect squares are RATIONAL

REVIEW OF NUMBER SYSTEMS:
Rational numbers are decimals that either terminate or repeat
which means they can be restated into a RATIO a/b of two integers a and b where b is not zero.
Natural numbers: 1, 2, 3, ... are RATIOnal because you can put them over 1
Whole numbers: 0, 1, 2, 3,....are RATIOnal because you can put them over 1
Integers: ....-3, -2, -1, 0, 1, 2, 3,....are RATIOnal because you can put them over 1
Rational numbers = natural, whole, integers PLUS all the bits and pieces in between that can be expressed as repeating or terminating decimals: 2/3, .6, -3.2, -10.7 bar, etc.
Real numbers: all of these! In Algebra II you will find out that there are Imaginary Numbers!
Square roots of NEGATIVE numbers are IMAGINARY

IRRATIONAL SQUARE ROOTS:
Square roots of a nonperfect squares are IRRATIONAL -
They cannot be stated as the ratio of two integers -
As decimals, they never terminate and never repeat -
you round them and use approximately sign.
MOST FAMOUS OF ALL IRRATIONAL NUMBERS IS PI!
(PI DAY IS THIS SATURDAY, BUT WE'LL CELEBRATE IT THIS NEXT MONDAY-- as we approximate PI DAY!)

There is another group of irrational numbers: Square roots of nonperfect squares
Square roots are MOSTLY IRRATIONAL!
There are fewer perfect squares than nonperfect!

Here are some perfect squares: 0, 4, 9, 16, 25, 36, etc.
PERFECT SQUARES CAN ALSO BE TERMINATING DECIMALS!
EXAMPLE: √( .04) is rational because it is ± .2

But all the square roots in between these perfect squares are IRRATIONAL
For example, the square root of 2, the square root of 3, the square root of 5, etc.

You can estimate irrational square roots.
For example, the SQRT of 50 is close to 7 because the square root of 49 is 7.
You can estimate that the SQ RT of 50 is 7.1 and then square 7.1 to see what you get.
If that's too much, try 7.05 and square that.
This works much better with a calculator!
And obviously, a calculator will give you irrational square roots to whatever place your calculator goes to.
Remember: These will never end or repeat
(even though your calculator only shows a certain number of places physically!)

RADICAL EXPRESSIONS 11-2
If an expression under the SQ RT sign is NEGATIVE, it does not exist in the REAL numbers!
There is no number that you can square and get a NEGATIVE PRODUCT

VARIABLES UNDER THE SQUARE ROOT SIGN:
If you have a variable under the SQ RT sign,
you need to determine what values of the variable will keep the radicand greater than or equal zero
The square root of x then is only real when x is greater than or equal to zero
The square root of (x + 2) is only real when x + 2 is greater than or equal to 0
Set x + 2 greater than or equal to 0 and solve as an inequality!
You will find that x must be greater than or equal to -2

SPECIAL CASE!!!! a variable squared plus a positive integer under radical:
If you're trying to find the principal square root of x2 (or any variable squared) plus a positive integer, then all numbers will work because a squared number will always end up either positive or zero!
Example: √(x² + 3) under the radical, any number positive or negative will keep the radicand positive (real), because once you square it, it is positive.
Then you're just adding another positive number.

If there is a variable squared and then a negative number (subtraction), the square will need to be equal or greater than that negative number to stay zero or positive under the radical.
EXAMPLE: √(x² - 10)
x² must be equal or greater than 10, so x must be at least the square root of 10
(the square root of 10 squared is 10)

ANOTHER SPECIAL CASE!!!!!!!!!!
ANY RADICAL EXPRESSION THAT HAS A VARIABLE SQUARED IS SIMPLIFIED TO THE ABSOLUTE VALUE OF THE VARIABLE.
Example: The square root of x² is the absolute value (positive) of x ( IxI )
Why?
Because it is assumed that you're finding the PRINCIPAL (positive) square root.
EXAMPLE:
x = -3
√x² = √(-3)² = √9 = 3 (not -3)
so you have to put absolute value signs around the answer
IF THERE IS A VARIABLE SQUARED
(see p. 489 #17-30)

TRINOMIALS UNDER THE RADICAL:
What do you think you would do if you saw x² + 10x + 25 under the radical sign????
FACTOR IT!
IT MAY BE A PERFECT SQUARE (a binomial squared!)

EXAMPLE:
√( x² + 10x + 25) factors to √(x + 5)² = I x + 5I