Rationalizing nominators?

Question

can someone please help me rationalize this?

( cube root of x ^2z ) divided by y

Answer 1

Remember that another way of writing [cubed root(x²z)] is (x²z)^(1/3).  You probably know that when you multiply powers with the same base, you add together their exponents.  For example,

x² * x³ = x^(2 + 3) = x^5

When you rationalize the numerator, you want (x²z) to be to the first power so that it will just be x²z.

So, (x²z)^(1/3) times (x²z) to what power gives x^1?

(x²z)^(1/3) * (x²z)^(?) = x^1.  

(1/3) + ? = 1

1 - 1/3 = ?

3/3 - 1/3 = 2/3

So (x²z)^(1/3) should be multiplied by (x²z)^(2/3) to get x²z.

(x²z)^(2/3) can also be written as cubed root[(x²z)²] OR

[cubed root(x²z)]².

Ok, now here is the actual computation:

[cubed root(x²z)]/y.  Multiply this by 1 in the form of

cubed root[(x²z)²] / cubed root[(x²z)²]:

([cubed root(x²z)] / y) * (cubed root[(x²z)²] / cubed root[(x²z)²]).  As a result, you'll get this:

(x²z) / (y * cubed root[(x²z)²]).  

I don't know if you want the answer simplified.  If so, then this is how it can be simplified:

= (x²z) / (y * cubed root[(x^4)z²])

= (x²z) / (y * xcubed root(xz²))

= (xz) / (y * cubed root(xz²))

Here is a link to the pdf file I made to show the work more clearly:

http://www.mediafire.com/file/fochswqdt5...

Hope this helps!