Algebra Period 3

Review Chapter 11: RADICALS

REMEMBER:
THE SQUARE ROOT OF ANYTHING SQUARED IS ITSELF!!!

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!)
x² + 10x + 25 factors to (x + 5)² so the square root of (x + 5)² = lx + 5l
Please read as absolute value of (x +5)

SIMPLIFYING RADICALS
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 )

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.

HELPFUL WAYS TO ATTACK SIMPLIFYING:
Factor trees, Inverted Division, Prime Factorization

Whenever there are 2 factors that are same, it can be simplified
(Take one of the 2 factors out of the radical and leave none under)
If there is already a factor outside the front of the radical, when you bring out another factor from underneath the radical, you multiply it with what was already outside in front.

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
We saw this already in factoring!!!
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)

Don't forget to always:
RATIONALIZE THE DENOMINATOR
The rule is that the radical is not simplified until all radicals are removed from the denominator
Simply use the equivalent fraction approach and multiply both the numerator and the denominator by the radical.

ADDING AND SUBTRACTING RADICALS:
Radicals function like variables, so you can only COMBINE LIKE RADICALS!

CHAPTER 11-7 (an old friend) - 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