SOLUTION: explain the first condition that must be met for a simplified radical. Explain why 5 divided by the square root of 2 is not simplified and demonstrate the steps we must take to si

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Question 273850: explain the first condition that must be met for a simplified radical. Explain why 5 divided by the square root of 2 is not simplified and demonstrate the steps we must take to simplify it.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
5%2Fsqrt%282%29
Simplified expressions with radicals:
  • Have no radicands (the expression inside the radical) which have factors (except 1) that are exact powers of the root. For example:
    • Square roots have no perfect square factors in the radicand.
    • Cube roots have no perfect cube factors in the radicand.
    • Fourth roots have no perfect 4th power factors in the radicand.
    • ....
  • Have no radicals in denominators.

Your expression has one radical. Its radicand, 2, has no perfect square factors (except 1) so it passes the first test. But the radical is in the denominator so it fails the second test. So your expression needs simplification. (Since simplified radicals are irrational, the process of eliminating radicals in the denominator is called "rationalizing the denominator".

To rationalize a one-term denominator like yours, you just multiply the numerator and denominator by whatever radical will make the radicand in the denominator a power of the root. Since your radical is a square root, we want the radicand in the denominator to be a perfect square. Square roots are the easiest to rationalize. To make your denominator's radicand a perfect square we just multiply the numerator and denominator by sqrt%282%29:
%285%2Fsqrt%282%29%29%28sqrt%282%29%2Fsqrt%282%29%29
which simplifies as follows:
5sqrt%282%29%2Fsqrt%284%29
5sqrt%282%29%2F2
This expression passes both tests so it is the simplified expression.

FWIW, if your expression had been 5%2Froot%283%2C+2%29 instead, we would have rationalized the denominator this way:
%285%2Froot%283%2C+2%29%29%28root%283%2C+2%5E2%29%2Froot%283%2C+2%5E2%29%29
5root%283%2C+4%29%2Froot%283%2C+2%5E3%29
5root%283%2C+4%29%2F2

Rationalizing denominators of more than one term, like 5%2F%28sqrt%282%29%2B1%29, is done differently because if we multiply the denominator by sqrt%282%29 we get 2+%2B+sqrt%282%29 which still has a radical in it. When you get a problem like this, post another question if you haven't learned how to handle them.