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