Pre Algebra Periods 1, 2, & 4

SQUARE ROOTS & PYTHAGOREAN THEOREM 11-1 & 11-2

Square root undoes squaring!
So you're looking for the number/variable that was squared to get the radicand
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 4. 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)
4) SQUARE ROOTS: 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

You can estimate nonperfect square roots by guess and check
Find the 2 numbers that it is between
Example: Square root of 52
It's between the 2 perfect squares: 49 and 64
So the square root is between 7 and 8
Since 52 is only 3 away from 49 and 12 away from 64, the square root will be closer to 7
Guess: 7.2 Square this: (7.2)(7.2) = 51.84 (adjust your estimate as necessary)

REAL NUMBER SYSTEM
2 PARTS: RATIONAL AND IRRATIONAL (both real)

Review LEAP FROG number systems
I: RATIONAL NUMBERS (definitions of different number systems):
Natural = counting = 1, 2, 3, . . .
Whole = natural + 0 = 0, 1, 2, 3, . . .
Integers = whole + opposites = -3, -2, -1, 0, 1, 2, 3, . . .
Rational = integers and all the fractions/decimals in between - terminating and repeating decimals

II: IRRATIONAL - numbers like pi and square root of 3 - never repeat or terminate - round!

PYTHAGOREAN THEOREM
FOR RIGHT TRIANGLES ONLY!
2 legs - make the right angle - called a and b
(doesn't matter which is which because you will add them and adding is COMMUTATIVE!)
hypotenuse - longest side across from the right angle - called c
You can find the third side of a right triangle as long as you know the other two sides:
a² + b² = c²
After squaring the two sides that you know, you'll need to find the square root of that number to find the length of the missing side (that's why it's in this chapter!)
EASIEST - FIND THE HYPOTENUSE (c)
Example #1 from p. 510
8² + 15² = c²
64 + 225 = c²
289 = c²
c = 17

A LITTLE HARDER - FIND A MISSING LEG (Either a or b)
Example #5 from p. 510
5² + b² = 13²
25 + b² = 169
b² = 169 - 25
b² = 144
b = 12

CONVERSE OF PYTHAGOREAN THEOREM
If you add the squares of the legs and that sum EQUALS the square of the longest side, it's a RIGHT TRIANGLE.
If you add the squares of the 2 smallest sides and that sum is GREATER THAN the square of the longest side, you have an ACUTE TRIANGLE.
If you add the squares of the 2 smallest sides and that sum is LESS THAN the square of the longest side, you have an OBTUSE TRIANGLE.