document.write( "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 \n" ); document.write( "

Algebra.Com's Answer #396320 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"sqrt%2848x%5E3y%5E5%29%2Fsqrt%2818x%5E5%29\"$
\n" ); document.write( "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:

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.
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.)
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.
Simplify any remaining square roots.

Let's see this in action:

\n" ); document.write( "1. Merge the square roots
\n" ); document.write( " $\"sqrt%2848x%5E3y%5E5%2F18x%5E5%29\"$

\n" ); document.write( "2. Reduce
\n" ); document.write( " $\"sqrt%288y%5E5%2F3x%5E2%29\"$

\n" ); document.write( "3.1 Make the denominator a perfect square:
\n" ); document.write( " $\"sqrt%28%288y%5E5%2F3x%5E2%29%283%2F3%29%29\"$
\n" ); document.write( " $\"sqrt%2824y%5E5%2F9x%5E2%29\"$

\n" ); document.write( "3.2 Split the square root
\n" ); document.write( " $\"sqrt%2824y%5E5%29%2Fsqrt%289x%5E2%29\"$

\n" ); document.write( "4. Simplify
\n" ); document.write( " $\"sqrt%284%2Ay%5E2%2Ay%5E2%2A6y%29%2F3x\"$
\n" ); document.write( " $\"%28sqrt%284%29%2Asqrt%28y%5E2%29%2Asqrt%28y%5E2%29%2Asqrt%286y%29%29%2F3x\"$
\n" ); document.write( " $\"%282%2Ay%2Ay%2Asqrt%286y%29%29%2F3x\"$
\n" ); document.write( " $\"%282y%5E2%2Asqrt%286y%29%29%2F3x\"$

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

\n" ); document.write( "2. Reduce
\n" ); document.write( "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:
\n" ); document.write( " $\"sqrt%2824y%5E5%2F9x%5E2%29\"$
\n" ); document.write( "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. \n" ); document.write( "

\n" );