SOLUTION: Please? Problem: Square root of: 7x^(19)y^(7) All divided by: The square root of: 2x^(2)y^(2) I'm thinking I should eliminate because they're not "families" but I just want t

Algebra ->  Square-cubic-other-roots -> SOLUTION: Please? Problem: Square root of: 7x^(19)y^(7) All divided by: The square root of: 2x^(2)y^(2) I'm thinking I should eliminate because they're not "families" but I just want t      Log On


   

Question 374497: Please? Problem:
Square root of: 7x^(19)y^(7)
All divided by:
The square root of: 2x^(2)y^(2)
I'm thinking I should eliminate because they're not "families" but I just want to be sure. Thanks!
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%287x%5E19%2Ay%5E7%29%2Fsqrt%282x%5E2%2Ay%5E2%29
Simplifying square roots involves making sure that
  • No square root has a perfect square factor
  • No square root has a fraction in it
  • No fractions have a square root in its denominator
When the expression does not involve fractions only the first item needs to be addressed.

Any valid math which achieves these goals will work. With an expression like yours, a fraction of square roots, what I like to do is:
  1. Use the property of radicals, root%28a%2C+p%29%2Froot%28a%2C+q%29+=+root%28a%2C+p%2Fq%29, to turn the expression into a square root of a fraction.
  2. Reduce the fraction as much as possible.
  3. If there is still a fraction after the reducing,
    1. turn its denominator into a perfect square.
    2. Use the same property as step #1, only in reverse, to turn the expression back into a fraction of square roots
    3. Simplify the denominator (which should be the square root of a perfect square)
  4. If there is still a square root at this point, use another property of radicals, root%28a%2C+p%29%2Aroot%28a%2C+q%29+=+root%28a%2C+p%2Aq%29, to separate out perfect square factors, if any.
Let's see how this works:
1) Square root of a fraction
sqrt%28%287x%5E19%2Ay%5E7%29%2F%282x%5E2%2Ay%5E2%29%29
2) Reduce the fraction
sqrt%28%287x%5E2%2Ax%5E17y%5E2%2Ay%5E5%29%2F%282x%5E2%2Ay%5E2%29%29

sqrt%28%287%2Ax%5E17%2Ay%5E5%29%2F2%29
3.1) Make the remaining denominator a perfect square:
sqrt%28%28%287%2Ax%5E17%2Ay%5E5%29%2F2%29%2A%282%2F2%29%29
sqrt%28%2814%2Ax%5E17%2Ay%5E5%29%2F4%29
3.2) Rewrite as a fraction of square roots:
sqrt%2814%2Ax%5E17%2Ay%5E5%29%2Fsqrt%284%29
3.3) Simplify the denominator
sqrt%2814%2Ax%5E17%2Ay%5E5%29%2F2
4) Simplify the remaining square root

(Use the Commutative property to rearrange the order so all the perfect squares are together.)


%28x%2Ax%2Ax%2Ax%2Ax%2Ax%2Ax%2Ax%2Ay%2Ay%2Asqrt%2814xy%29%29%2F2
%28x%5E8%2Ay%5E2%2Asqrt%2814xy%29%29%2F2
This is the simplified expression. (It meets all three conditions described above.)

Note: Square roots, unless they have a "-" in front of them, are supposed to be positive (or zero). When simplifying an expression of square roots, you sometimes need to use absolute value to ensure your simplified expression is positive (or zero), too. In this problem, however, the x%5E8%2Ay%5E2 cannot be negative because of the even exponents. So we do not need any absolute values.