Radicals

By Yusuf Mohamed on Saturday, October 19, 2002 - 04:19 am:

I have a big problem with radicals.
So what in the world is 7 sqrt 3 suppose to mean?
And how are the ways to put it back to normal?

Thanks alot!!

By Marcos Charalambides on Saturday, October 19, 2002 - 06:14 am:

Okay I'll try explain the general idea...

If you have the equation x2 = 4 (x is positive) the solution is simply 2
But what if you had x2 = 2?
Using a calculator you get 1.41421... but this is only an approximation so it is better if you just leave the answer as sqrt 2. This is more concise and more accurate as you're not rounding off anything...

Now,
Using the fact that (ab)n = an bn and noting that p1/2 means the square root of p we can do the following:
Say we had 12
We can write this as (12)1/2 = (3x4)1/2
By the rule I mentioned before you can split this up...
Therefore,
(3x4)1/2 = (3)1/2 (4)1/2
But 41/2 =2 and so (3 )1/2 (4 )1/2 =3(2) (or 23, it's more common to put the rational number first)
So we have...
12=23 ( 23 basically just means 2 multiplied by 3)
Can you now see that 8=22 and 45=35...

So, in response to your original question:
73 means 7 multiplied by 3...

Can you see (by reversing the process above) why this is ?

So we can write this as:

  1. 73
  2. 147
  3. 12.1243556...

"And how are the ways to put it back to normal?"
Now can you see why (1) is preferable (it's shorter and there are simpler numbers)?

Now can you see what the answer is to your geometric problem?

If there's anything you didn't understand or I didn't explain well please ask...