document.write( "Question 398823: Please help, I am soooo lost with 2 problems:
\n" ); document.write( "1: Cube Root of 270x divided by cube root of 10xy^2\r
\n" ); document.write( "\n" ); document.write( "2: Cube root of x^2 divided by cube root of 4\r
\n" ); document.write( "\n" ); document.write( "Thank you so much! \n" ); document.write( "

Algebra.Com's Answer #282507 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
With expressions like these, a radical over the same kind of radical, I like to use the following procedure:

Use the property $\"root%28a%2C+p%29%2Froot%28a%2C+q%29+=+root%28a%2C+p%2Fq%29\"$ to combine the two radicals into a single radical.
Reduce the fraction inside the radical, if possible.
If the denominator is not a perfect power of the type of radical, then multiply the numerator and denominator by some expression so that the denominator becomes a perfect power of the type of radical.
Use the same property as step 1, only in reverse, to split the radical back to a radical over a radical.
Simplify. If the previous steps were done correctly, there should be no radicals remaining in the denominator.

\n" ); document.write( "Let's see this in action.
\n" ); document.write( " $\"root%283%2C+170x%29%2Froot%283%2C+10xy%5E2%29\"$
\n" ); document.write( "1) Combine radicals:
\n" ); document.write( " $\"root%283%2C+170x%2F10xy%5E2%29\"$
\n" ); document.write( "2) Reduce the fraction inside. The x's cancel and a factor of 10 cancels leaving:
\n" ); document.write( " $\"root%283%2C+17%2Fy%5E2%29\"$
\n" ); document.write( "3) If the denominator is is not a perfect power of the type of radcial, then multiply the numerator and denominator by whatever makes the denominator a perfect power of the type of radical. Since we are working with cube roots, we are looking to make the denominator a perfect cube. All we need is another factor of y:
\n" ); document.write( " $\"root%283%2C+%2817%2Fy%5E2%29%28y%2Fy%29%29\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"root%283%2C+17y%2Fy%5E3%29\"$
\n" ); document.write( "4) Split the radical:
\n" ); document.write( " $\"root%283%2C+17y%29%2Froot%283%2C+y%5E3%29\"$
\n" ); document.write( "5. Simplify.
\n" ); document.write( " $\"root%283%2C+17y%29%2Fy\"$
\n" ); document.write( "Since there are no perfect cube factors in 17y, the radical in the numerator will not simplify further.

\n" ); document.write( " $\"root%283%2C+x%5E2%29%2Froot%283+4%29\"$
\n" ); document.write( "1. Combine the radicals:
\n" ); document.write( " $\"root%283%2C+x%5E2%2F4%29\"$
\n" ); document.write( "2. Reduce the fraction. This fraction will not reduce.
\n" ); document.write( "3. Make the denominator a power of the type of radical. The \"nearest\" perfect cube to 4 is 8. So we just have to multiply the numerator and denominator by 2:
\n" ); document.write( " $\"root%283%2C+%28x%5E2%2F4%29%282%2F2%29%29\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"root%283%2C+2x%5E2%2F8%29\"$
\n" ); document.write( "4. Split the radical:
\n" ); document.write( " $\"root%283%2C+2x%5E2%29%2Froot%283%2C+8%29\"$
\n" ); document.write( "5. Simplify.
\n" ); document.write( " $\"root%283%2C+2x%5E2%29%2F2\"$
\n" ); document.write( "There are no perfect cube factors in $\"2x%5E2\"$ so the numerator will no simplify any further.

\n" ); document.write( "Note 1: The procedure I've described is not the only way to simplify your expressions. But it is pretty efficient at handing radical over radical expression.
\n" ); document.write( "Note 2: Any procedure used correctly should simplify your expressions to the same results as we got above.
\n" ); document.write( "Note 3: If the radicals are of different types then use fractional exponents to make them the same type. For example:
\n" ); document.write( " $\"sqrt%282x%29%2Froot%283%2C+y%5E2%29\"$
\n" ); document.write( " $\"%282x%29%5E%281%2F2%29%2F%28y%5E2%29%5E%281%2F3%29\"$
\n" ); document.write( " $\"%282x%29%5E%283%2F6%29%2F%28y%5E2%29%5E%282%2F6%29\"$
\n" ); document.write( " $\"%28%282x%29%5E3%29%5E%281%2F6%29%2F%28%28y%5E2%29%5E2%29%5E%281%2F6%29\"$
\n" ); document.write( " $\"root%286%2C+%282x%29%5E3%29%2Froot%286%2C+%28y%5E2%29%5E2%29\"$
\n" ); document.write( " $\"root%286%2C+8x%5E3%29%2Froot%286%2C+y%5E4%29\"$
\n" ); document.write( "Now that the radicals are both 6th roots, we can use the procedure above:
\n" ); document.write( "1. Combine:
\n" ); document.write( " $\"root%286%2C+8x%5E3%2Fy%5E4%29\"$
\n" ); document.write( "2. Reduce. This will not reduce.
\n" ); document.write( "3. Make the denominator a power of the type of radical:
\n" ); document.write( " $\"root%286%2C+%288x%5E3%2Fy%5E4%29%28y%5E2%2Fy%5E2%29%29\"$
\n" ); document.write( " $\"root%286%2C+%288x%5E3y%5E2%29%2Fy%5E6%29\"$
\n" ); document.write( "4. Split
\n" ); document.write( " $\"root%286%2C+8x%5E3y%5E2%29%2Froot%286%2C+y%5E6%29\"$
\n" ); document.write( "5. Simplify. Since sixth roots are supposed to be positive and since we do not know if y is positive, we should use absolute value when simplifying $\"root%286%2C+y%5E6%29\"$ :
\n" ); document.write( " $\"root%286%2C+8x%5E3y%5E2%29%2Fabs%28y%29\"$ \n" ); document.write( "

\n" );