Help with a Pre-calculus Problem.?

Question

Ok, Keep getting the worng answer for the denominator. I have tried everything, and after about a week, I have just decided to ask my yahoo! buddies.

Simplify:
http://i122.photobucket.com/albums/o259/RandyQuaid/RADICAL_problem1.gif

I know the right answer, I just don't know why the denominator is what it is. Please explain mainly why the denominator reduces the way it does. THANKS!

Answer 1

I get:

x * fourthroot(15x²y³)
----------------------------
3

Is that the right answer?

If so, I got it by first rewriting the problem as:

fourthroot(5 * x^8 * y*3)
------------------------------
fourthroot(27x^2)

Then I factored out the x^8 as x², and then multiplied the top and bottom by:

fourthroot(3x²)

This gave me the following result:

x² * fourthroot(15x²y³)
-------------------------------
fourthroot(81x^4)

x² * forthroot(15x²y³)
------------------------------ { simplify the denominator }
3x

x * fourthroot(15x²y³)
------------------------------- { cancel out the x on top and bottom }
3

Answer 2

The denominator would just be 27, since the x^2 divides into the x^8.

Answer 3

A root of a fraction is only considered simplified when there is no radical in the denominator. Let's look at the fraction:

4rt((5x^8y^3)/27x^2)

The x^2 cancels out of the denominator by dividing it into the x^8:

4rt((5x^8y^3)/27x^2) = 4rt((5x^6y^3)/27)

Now, 27=3^3, so if we multiply both numerator and denominator by 3, you'll have a fourth power in the denominator, which can then be simplified:

4rt((5x^6y^3)/27) = 4rt((15x^6y^3)/3^4)
=4rt(15x^6y^3)/3 = (x/3)4rt(15x^2y^3)

Notice that the denominator is simply 3 without a radical, and thus, it's now in simplest form.

Answer 4

[ (5x^8y^3) / (27x^2) ]^1/4

Distribute the 1/4 exponent.
(5^1/4) (x^2) (y^3/4)(27^-1/4) (x^-2/4)

Combine like terms.
(5^1/4) (y^3/4) (27^-1/4) (x^3/2)

You didn't say in what form you wanted the answer. The answer above is fine. If you want it in fraction form, then convert to a fraction.

[ (5^1/4) (y^3/4) (x^3/2) ] / (27^1/4)

Rationalize the denominator.

[ (5^1/4) (y^3/4) (x^3/2) ] / (27^1/4) MULTIPLIED BY (3^1/4) / (3^1/4)

[ (15^1/4) (y^3/4) (x^3/2) ] / (81^1/4)

[ (15^1/4) (y^3/4) (x^3/2) ] / 3

Answer 5

27 = 3^3. 3^3^(1/4)=3^(3/4)

x^2^(1/4) = √x