Algebra Period 3 (Wednesday)

PRODUCT OF RADICALS 11-4
Basically, a radical is similar to a variable in that you can always multiply them,
but only add or subtract them if they are the exact same radicand (like terms)

√2 •√14 = √28
√28 can be simplified to 2√7, read 2 "rad" 7

HELPFUL HINT:
If you are multiplying √250 • √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:
√250 • √50
Factor each number first:
√(2 x 5 x 5 x 5) •√(2 x 5 x 5)
Combine under one radical sign in PAIRS:
√(2 x 2)(5 x 5)(5 x 5)(5)
Simplify by taking one of each pair out of the radical
(2 x 5 x 5)√5
Multiply all the perfect roots that you took out of the radical:
50√5

Dividing & Simplifying 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!
√(3/16)
Notice that the denominator is a perfect square so it makes sense to look at the denominator separately from the numerator:
√3/√16 = √3/4


√ (25/36)
Notice that both the numerator and denominator are perfect squares so it makes sense to simplify them apart:
√25/√36 = 5/6


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: √(27/3)
27/3 simplifies to 9:
√9 = 3

Notice that if you took this fraction apart first and
then tried to find the square root of each part, it's much more complicated:
√(27/3)
Separate the numerator from the denominator:
√27 / √ 3

Factor 27:
√(3x3)x3
√3
Simplify the numerator in pairs:
3√3/√3

Cross cancel if possible:
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:
√7/√ 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 √3 in the denominator.

You need to multiply both numerator and denominator by √3 to get it out of there:
√7 /√ 3 = √7 • √3 /√ 3•√ 3 = √(7• 3) /3 = √21 /3

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!)
The √21 is not 21!
It's irrational and approximately 4.58
You can't cross cancel 4.58 with 3 in the denominator!

Math 6 H Periods 1, 6 & 7

Addition and Subtraction of Fractions 7-1

Most of you already know how to add and subtract fractions, although some of you may need just a little review.

5/9 + 2/9 = 7/9

and that
7/9 – 2/9 = 5/9


The properties of addition and subtraction of whole numbers also apply to fractions.
If the denominators are the same— add or subtract the numerators AND use the numerator!!

In order to add two fractions with different denominators, we first find two fractions, with a common denominator, equivalent to the given fractions. Then add these two fractions.

The most convenient denominator to use as a common denominator is the least common denominator of LCD, of the two fractions. That is, the least common multiple of the two denominators.

LCD ( a/b, c/d) = LCM(b, d) where band d both cannot be equal to 0

For example LCD ( 3/4, 5/6) = LCM(4,6) =12

3/4 = 9/12 and 5/6 = 10/12

Let’s do:
7/15 + 8/9

First find the LCD

LCM(15, 9) Do your factor trees or inverted division – or just by knowing!!

15 = 3• 5
9 = 32

So LCM(15,9) = [every factor to its greatest power] 32•5 = 45

Then find equivalent factions with a LCD of 45, and add

7/15 = 21/45

8/9 = 40/45

21/45 + 40/45 = 61/45 = 1 16/45


Addition and Subtraction of Mixed Numbers 7-2

To add or subtract mixed numbers we could first change the mixed numbers to improper fractions and then use the method from 7-1 .
1 4/9 + 3 1/9 = 13/9 + 28/9 = 41/9 = 4 5/9 but that was 5th grade….
In the second method, and the one I prefer, you work separately with the fractional and whole number parts of the given mixed numbers.


STACK THEM!!
3 4/9
1 7/9
4 11/9 = 5 2/9


If the fractional parts of the given mixed numbers have different denominators, we find equivalent mixed numbers whose fractional parts have the same denominator, usually the LCD.

5 3/10 + 7 7/15

Stack

5 3/10
+7 7/15

Draw a line separating the fractional part from the whole numbers Find the LCM of the denominators the LCD and add…

9 5/9 - 4 13/15

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

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