Algebra (Period 4)

SIMPLIFYING RADICALS 11-3

SIMPLIFYING NONPERFECT NUMBERS UNDER THE RADICAL:
A simplified radical expression is one where there is no perfect squares left under the radical sign
You can factor the expression under the radical to find any perfect squares in the number:
EXAMPLE: square root of 50 = SQRT(25 * 2)
Next, simplify the SQRT of the perfect square and leave the nonperfect factor under the radical:
SQRT(25 * 2) = SQRT(25) *SQRT(2) = 5 SQRT( 2 )
***Imagine the square root sign top extends over all numbers****
√(25 * 2) = √25 * √2 = 5√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?
You can even use all the perfect squares that you have memorized... it really helps// just remember when you pull out a number... you are pulling out the number which multiples by itself to get that perfect square... NOT the perfect square itself...


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: SQRT 250
Prime factorization = 2 * 5 * 5 * 5
Circle the first two 5's
5 * 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: SQRT 250 = 5 SQRT 10
√250 = 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: SQRT (x¹⁰ ) = x⁵ or
√x¹⁰ = x⁵
We saw this already in factoring!!!


SIMPLIFYING NONPERFECT VARIABLES:
If the variable has an even power:
EXAMPLE: The square root of 75x¹⁰ = SQRT [(25)(3)(x10 )] =
SQRT [ (5)(5)](3)(x⁵x⁵)

***Imagine the square root sign top extends over all numbers****
√75x¹⁰ = √[(25)(3)(x¹⁰ )] = √[ (5)(5)](3)(x⁵x⁵)


Simplified, you can pull out a factor of 5 and x⁵
Final simplified radical = 5x⁵ SQRT(3)
or
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 that odd power times that variable to the 1 power:
Example: x⁵ = x⁴ x
so if you have the SQRT( x⁵ ) = SQRT (x⁴ x) = x²SQRT(x)

√x⁵ =√x⁴ x = x²√x



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: SQRT(3x² + 12x + 12) or

***Imagine the square root sign top extends over all numbers****
√3x² + 12x + 12

First factor out a 3:

SQRT [3 (x² + 4x + 4) ] or
***Keep imagining that line across all the numbers under the radical √ ****
√3(x² + 4x + 4)

Now factor the trinomial: SQRT [3 (x + 2) (x + 2) ] or

√3(x + 2)²

The (x + 2)² is a perfect square so SQRT [3 (x + 2) (x + 2) ] = (x + 2) SQRT( 3 )

or (x+2)√3

you need to include the HUGS around the (x +2) because you want that binomial to be multiplied by the √3 .. not just the 2.

Math 6H ( Periods 3, 6, & 7)

The Tangram Story
(One of the activities we did while students were at Astro Camp)

Once upon a time, long, long ago, in a faraway magical land lived a little boy named Tan. The emperor of the land had entrusted Tan with a very special task. The emperor had given Tan a magical square tile and asked him to deliver it to one of the emperor’s subjects who lived in the countryside. Tan was instructed to go directly to the subject’s house and not stop along the way. But, along the way, Tan encountered a group of his friends playing along the river. Tan thought he would stop for just a few minutes to rest and play … when alas; the tile flew out of his pocket and broke into seven pieces. Tan and his friends were so upset that they tried for days to put the tile back together again. They were able to form many beautiful designs of birds and animals and flowers but never the square tile. His designs have been handed down form generation to generation for over 3000 years and are known as Tangram puzzles.
If you would like to enjoy creating some of these puzzles or are interested in how to create your own tile, come in before or after school.

Algebra (Period 4)

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: √(x ² + 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 ( lxl )
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 = l x + 5l

Algebra (Period 4)

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 = 25½ = 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 on a Sunday this year, BUT we will celebrate it either Friday before or the Monday after... we will be approximating 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!)

Another way to calculate the square root.
Take the perfect square above and below.
Find the difference between the two perfect squares ( it will always bee their sum)
Use that number as the denominator.
Now take the difference between your number and the bottom perfect quare. That difference will be the numerator. Now, if necessary estimate-- using benchmarks to change your number into a decimal.

For instance, what is the square root so 181
Set up your problem like this

198
181
169

because 196 and 169 are your perfect squares between 181

Then you know
√196 = 14
181
√169 = 13

put the approximate sign after your number within a square root sign and place the bottom perfect square and a decimal point after that because you know your number falls between the two square roots., that is,

√196 = 14
√181 ≈ 13.
√169 = 13

Now, 196 - 169 = 27 but it is also try that 14 + 13 = 27... either way you use that number as the denominator.
The numerator is the difference between the number you are using and th bottom perfect square... in this case 181- 169 = 12
now 12/27 becomes 4/9 which we know is .44444... so
√181 ≈ 13.44