SOLUTION: The square root of x raised to the 3rd power divided by the square root of 12

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Question 404197: The square root of x raised to the 3rd power divided by the square root of 12
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!

While there are other ways, the way I like to simplify radical over radical fractions is:
  1. Use a property of radicals, , to rewrite the fraction of radicals into the radical of a fraction.
  2. Reduce the fraction inside the radical, if possible.
  3. 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.)
  4. Use the property again, this time from right to left, to rewrite the radical of a fraction as a fraction of radicals.
  5. Simplify the two radicals. If step 3 was done correctly then the radical in the denominator should disappear.

Let's see this in action.
1) Use the property to combine the radicals:

2) Reduce the fraction inside the radical, if possible.
This fraction, , will not reduce.
3) Multiply the numerator and denominator so that the denominator inside the radical becomes a power of the type of root.
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.

giving us:

4) Use the property to split the radicals:

5) Simplify the radicals.
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:




This is the simplified answer.

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:










We get the same answer. But did you notice the extra simplifying?
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