Expert: Josh - 4/25/2007

Question
My problem is 12 third root 5 over third root 5 + third root 2.  I tried to pull out the 12 and put third root 5 over 5+2 and came up with 12 third root 5 over third root 7.  Then I multiplied top and bottom by third root 7.  Am I doing this right?

Answer
Hi Angela,

According to your description, the expression is probably [12*5^(1/3)]/[5^(1/3)+2^(1/3)], where x^(1/3) represents the "third root of x". However, I don't know what the question actually asks for.

It seems to me that the cubic root is causing you some grief. One thing I can say is that the denominator 5^(1/3)+2^(1/3) CANNOT be rewritten as (5+2)^(1/3) or 7^(1/3) which I guess is what you are unsure about.

To simplify [12*5^(1/3)]/[5^(1/3)+2^(1/3)], we can divide both numerator and denominator by 5^(1/3).

For the numerator: 12*5^(1/3)/5^(1/3)=12.
For the denominator: [5^(1/3)+2^(1/3)]/5^(1/3) becomes 1 + (2/5)^(1/3). To see this, observe that the second term is third root of 2, divided by the third root of 5.

Because the same exponent (power law) is involved, we can in general rewrite x^a/y^a as (x/y)^a. In this problem, x=2, y=5 and the third root is equivalent to taking the exponent a=1/3.

So, we can simplify the expression as 12/[1+(2/5)^(1/3)]. This reads "twelve over (in bracket) one plus the third root of two fifths".