Warm up 1-8-12 week 3-1 Break the following down in to factors. If there are a common pair of factors at the end, circle them. If the factor does not have a match, put a square around it. 1. 80 2. 76 3. 64 4. 120 5. 20 6. 24

Simplifying Radical SWBAT: Multiply and simplify radical expressions containing both numbers and variables. Write down the first 20 perfect squares like this 12 = 1 22 = 4

The parts of a radical The index. The number above the “tail.” Normally it is a 2 and isn’t even written. The radical bar. Kind of like a division sign with a tail The Radicand. The “stuff” underneath the radical bar, can be numbers or variables.

How to Simplify Radicals The radical bar is kind of like “math” prison. Anyone that can, wants to get out of prison. But there are guards, so in order to bust out of the joint the numbers always break out in groups. The size of the group depends on the number of guards.

Step 1: Figure out the number of “guards” in the prison Step 1: Figure out the number of “guards” in the prison. The index is the number of guards. How many guards are there in the following problems?

Step 2: Figure out how many prisoners (the radicand) are going to escape. The number of guards determine the groups trying to escape. For example: has how many guards? 2 How many prisoners are there? 3

The Great Escape So here is how it goes down. Those 3 prisoners, not very smart, see that there are only two guards and think, hey, two of us can charge the guards and get away. So two of the prisoners go charging at the guards…but, only one gets away. One gets caught and sent to solitary confinement and is never seen of or heard from again. And the third guy, since he didn’t have a partner to charge the guards with, he’s left to rot in prison. That’s what happens when you lead a life of crime.

So… which is really just… so it ends up like this… This guy didn’t have a partner and so is left in prison This guy gets away This guy gets caught and is never heard from again But teacher, teacher, what happens to the other guy? He’s Dead, gone, went bye bye, won’t see him again! The guy that got away The guy left in prison

Here’s another example. How many guards? 2 How many prisoners? 5 How many are going to try to bust out? 4 Why only 4? The 5th guy doesn’t have a partner. How many actually make it outside if 4 try to escape? 2 How many are left in prison? 1, he didn’t have a partner. So then the simplified form of is…

Another Example Simplify the following radical expression. How many guards? How many prisoners are going to try to escape? How many left inside to rot? How many make it out of the big house? What is the answer?

Try these by yourself. They all follow the same steps. 1. 2. Little different, how many “guards” 3. 4.

What about numbers? What do you do if there are numbers underneath the radical instead of variables? Exactly the same thing…except You must break down the number into a “factor tree.”

Simplifying Radicals with Numbers. Simplify the following. Step one, break the number down into a factor tree. Step two: Circle any pairs. The pairs bust out of the klink just like normal. 2 10 2 5

Only one makes it out, if you didn’t have a partner to escape with, then you rot. So the answer is… 2 10 2 5

Another Example Simplify the following: Step 1: Make a factor tree So that leaves: Which is: 14 10 2 7 2 5

You Try Simplify the following: 5. 6. 7. 8.

Try Some Difficult Ones. Same rules (steps)! 9. 10. 11. 12. Put on your notes – this is a continuation of yesterday’s stuff

Multiplying and Dividing Radicals The rules to multiply and divide radicals are: You can multiply and divide radicals with other radicals Multiply or divide the terms outside Multiply or divide the terms inside the radical Simplify completely

Multiply the radicals then simplify Example :

Last thing you need to know about radicals. Last thing: You may not leave a radical in a denominator! Catch that, no radicals in the bottom of a fraction! If there is a radical in the denominator, you have two options.

The Two Options Are… Option A: Check, maybe you can simplify with the numerator. Example: Option B: Called RATIONALIZING Multiply the numerator and denominator by the radical in the denominator.

Last Ones, I Swear 13. 14. 15. 16.

Summary: To simplify radicals I must … Classwork: simplify the following 8 problems (put it on the back of your notes) Homework: Radical Worksheet