```B
7-2B:
Objective:
• To divide and simplify expressions with radicals.
If and are real numbers, then
(The only exception is if b = 0... then we cannot divide by 0!)
To simplify radicals with fractions as radicands, we NEVER want to leave a radical in the denominator!
When we simplify radicals with fractions (division), there are 3 things we need to ask:
EASY: (1) Is the denominator a "perfect" root?
MEDIUM: (2) Can numerator and denominator divide nicely?
HARD: (3) Rationalize the denominator (Get the hippie out of the basement!)
EASY: (1) Is the denominator a "perfect" root?
EX #1:
EX #2:
EX #3:
"outsides" can be simplified with "outsides"
It's ok to have a radical in the numerator, just not in the denominator.
MEDIUM: (2) Can numerator and denominator divide nicely?
EX #4:
EX #5:
Practice Problems:
Divide and simplify. Assume that all variables are positive.
1.
2.
3.
256 is a perfect fourth on the powers table (on the back of your 7­1 notes)
Instead of writing out x * x * x... 14 times, divide the exponent by the index. The whole number will be how many variables will "break out", the remainder will be how many variables are "stuck underneath".
HARD: (3) Rationalize the denominator (Get the hippie out of the basement!)
Go through all of the other questions first: • is the denominator a perfect root? if so, do it
• can the fraction be simplified? if so, do it
• then, if there is still a radical in the bottom, rationalize
EX #6:
The denominator is not a perfect square and 2/3 cannot be simplified, so we have to "force" the denominator to become a perfect number!
What we are multiplying the original fraction by is another fraction that is equal to 1, that's why we can do this...
We are choosing to multiply by the square root of 3 (over itself) because we want the 3 under the radical in the denominator to be able to "break out" from under the radical.
We do not want any radicals left in the denominator.
EX #7:
EX #8:
Here, we are choosing to multiply by two more 3's and two more x's because we need to make three of a kind under a cubed root. What we choose to multiply should be the least amount of numbers and variables possible to break everything out in the denominator.
Practice Problems:
Rationalize the denominator. Assume that all variables are positive.
4.
5.
simplify the fraction first
then rationalize
We cannot simplify the 5y in the top with the 5y in the bottom because the one in the top is inside the radical, the one in the bottom is outside of the radical... outsides and insides don't mix!
6.
simplify the fraction first
then rationalize
We could have chosen to multiply by 3 * 3 * x2 * x2 , but we will need to do more simplifying at the end of the problem. Try to select the least amount of numbers and variables to make the denominator perfect.
Application:
The distance d in meters that an object will fall in t seconds is given by: d = 4.9t2.
a. Express t in terms of d (solve for t). b. Rationalize the denominator.