Question 6996: the simplified radical of 84
Answer by prince_abubu(198)   (Show Source):
You can put this solution on YOUR website! This might sound really stupid, but at least you'll remember it. Think of the radical as a house where the people inside are being hooked up together. Once hooked up, they leave the house as "one couple". Like in real life, not everybody gets hooked up, so at the end, there will be the lonely singles inside, who will form a friendship instead of a "couple" relationship.
Let's take the 84. Actually, it's inside the house so  . We'll have to split up the 84 into a product of prime numbers. (So, a prime number actually represents a person).
Since 84 is divisible by 2 because it's even, we can rewrite the 84 as 2*42. Now, we have the radical rewritten as  . The 2 is already factored out, just like a single person might temporary leave a gang hanging out in one part of the house.
42 is still even, so, it's still divisible by 2. So we can say that 42 = 2 * 21. Plugging in the 2 * 21 in place of the 42 gives us  AHA! We've two 2's inside. We've got a compatible couple! They can how leave the house as 1 couple. Actually, the 2*2 "exits" the radical as 2. So now we've got 
There's a 21 left inside the radical. We know that we can split that up as 3*7, and that's it. Since 3 and 7 are both prime, they have no other factors but themselves and 1. So, while the two people (since the numbers are different) are incompatible, they just have to stay inside the house as friends.
Yes, you may leave the number 21 written as it is. You don't have to write them as a product of 7*3. After all, all you're doing is finding the couples and taking them outside the house. Whoever is left inside can just form their own gang (multiplied together) of single people.
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