SOLUTION: square root of 48 x to the power of 3 y to the power of 5 over the square root of 18 x to the power of 5

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Question 629523: square root of 48 x to the power of 3 y to the power of 5 over the square root of 18 x to the power of 5
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%2848x%5E3y%5E5%29%2Fsqrt%2818x%5E5%29
While there are manyu ways to to do, the way I like to simplify expressions like this, a square root over a square root, is:
  1. Use the root%28a%2C+p%29%2Froot%28a%2C+q%29+=root%28a%2C+p%2Fq%29 property of radicals change the fraction of square roots into the square root of a fraction.
  2. Reduce the fraction. (If you're clever and you know what the next step is, then you may choose not to reduce the fraction fully. I'll show you what I mean later.)
  3. If there is still a denominator, then...
    1. Make it a perfect square.
    2. Use the roott%28a%2C+p%29%2Froot%28a%2C+q%29+=root%28a%2C+p%2Fq%29 property again, this time in reverse, to create a fraction of square roots.
  4. Simplify any remaining square roots.
Let's see this in action:

1. Merge the square roots
sqrt%2848x%5E3y%5E5%2F18x%5E5%29

2. Reduce
sqrt%288y%5E5%2F3x%5E2%29

3.1 Make the denominator a perfect square:
sqrt%28%288y%5E5%2F3x%5E2%29%283%2F3%29%29
sqrt%2824y%5E5%2F9x%5E2%29

3.2 Split the square root
sqrt%2824y%5E5%29%2Fsqrt%289x%5E2%29

4. Simplify
sqrt%284%2Ay%5E2%2Ay%5E2%2A6y%29%2F3x
%28sqrt%284%29%2Asqrt%28y%5E2%29%2Asqrt%28y%5E2%29%2Asqrt%286y%29%29%2F3x
%282%2Ay%2Ay%2Asqrt%286y%29%29%2F3x
%282y%5E2%2Asqrt%286y%29%29%2F3x

P.S. Now I'm going to repeat the first two steps using "clever" reducing:
1. Merge the square roots
sqrt%2848x%5E3y%5E5%2F18x%5E5%29

2. Reduce
I'm still going to reduce the x's as before. But instead of dividing the numerator and denominator by 6 like I did above, I am going to just divide by 2:
sqrt%2824y%5E5%2F9x%5E2%29
The fraction is not fully reduced, But look at the denominator! It is now a perfect square. So by thinking ahead we can skip step 3.1.