document.write( "Question 404197: The square root of x raised to the 3rd power divided by the square root of 12 \n" ); document.write( "

Algebra.Com's Answer #285883 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"sqrt%28x%5E3%29%2Fsqrt%2812%29\"$
\n" ); document.write( "While there are other ways, the way I like to simplify radical over radical fractions is:

Use a property of radicals, $\"root%28a%2C+p%29%2Froot%28a%2C+q%29+=+root%28a%2C+p%2Fq%29\"$ , to rewrite the fraction of radicals into the radical of a fraction.
Reduce the fraction inside the radical, if possible.
Multiply the numerator and denominator of the fraction by the same expression so that the denominator a power of the type of root. (Since you have square roots we will be making the denominator a perfect square. If they had been cube roots, we would make the denominator a perfect cube. If they had been 4th roots, etc.)
Use the $\"root%28a%2C+p%29%2Froot%28a%2C+q%29+=+root%28a%2C+p%2Fq%29\"$ property again, this time from right to left, to rewrite the radical of a fraction as a fraction of radicals.
Simplify the two radicals. If step 3 was done correctly then the radical in the denominator should disappear.

\n" ); document.write( "Let's see this in action.
\n" ); document.write( "1) Use the property to combine the radicals:
\n" ); document.write( " $\"sqrt%28x%5E3%2F12%29\"$
\n" ); document.write( "2) Reduce the fraction inside the radical, if possible.
\n" ); document.write( "This fraction, $\"x%5E3%2F12\"$ , will not reduce.
\n" ); document.write( "3) Multiply the numerator and denominator so that the denominator inside the radical becomes a power of the type of root.
\n" ); document.write( "In this expression we will make the denominator into a perfect square. The obvious choice would be to multiply the numerator and denominator by 12. But we can do better: Multiplying the numerator and denominator by 3. This will make the denominator a 36 which is a perfect square. The lower perfect square saves us from extra simplifying later on.
\n" ); document.write( " $\"sqrt%28%28x%5E3%2F12%29%283%2F3%29%29\"$
\n" ); document.write( "giving us:
\n" ); document.write( " $\"sqrt%283x%5E3%2F36%29\"$
\n" ); document.write( "4) Use the property to split the radicals:
\n" ); document.write( " $\"sqrt%283x%5E3%29%2Fsqrt%2836%29\"$
\n" ); document.write( "5) Simplify the radicals.
\n" ); document.write( "The denominator, being the square root of a perfect square, simplifies to a nice whole number. The numerator is not a perfect square. But it does have a perfect square factor:
\n" ); document.write( " $\"sqrt%283x%5E2%2Ax%29%2F6\"$
\n" ); document.write( " $\"sqrt%28x%5E2%2A3x%29%2F6\"$
\n" ); document.write( " $\"%28sqrt%28x%5E2%29%2Asqrt%283x%29%29%2F6\"$
\n" ); document.write( " $\"%28x%2Asqrt%283x%29%29%2F6\"$
\n" ); document.write( "This is the simplified answer.

\n" ); document.write( "When we were making the denominator a perfect square we turned it into a 36. If we had used the more obvious choice, here is how it would have worked out:
\n" ); document.write( " $\"sqrt%28%28x%5E3%2F12%29%2812%2F12%29%29\"$
\n" ); document.write( " $\"sqrt%28%2812x%5E3%29%2F144%29\"$
\n" ); document.write( " $\"sqrt%2812x%5E3%29%2Fsqrt%28144%29\"$
\n" ); document.write( " $\"sqrt%284%2A3%2Ax%5E2%2Ax%29%2F12\"$
\n" ); document.write( " $\"sqrt%284%2Ax%5E2%2A3%2Ax%29%2F12\"$
\n" ); document.write( " $\"%28sqrt%284%29%2Asqrt%28x%5E2%29%2Asqrt%283%2Ax%29%29%2F12\"$
\n" ); document.write( " $\"%282%2Ax%2Asqrt%283%2Ax%29%29%2F12\"$
\n" ); document.write( " $\"%282%2Ax%2Asqrt%283%2Ax%29%29%2F%282%2A6%29\"$
\n" ); document.write( " $\"%28cross%282%29%2Ax%2Asqrt%283%2Ax%29%29%2F%28cross%282%29%2A6%29\"$
\n" ); document.write( " $\"%28x%2Asqrt%283%2Ax%29%29%2F6\"$
\n" ); document.write( "We get the same answer. But did you notice the extra simplifying? \n" ); document.write( "

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