Find ab given its relationship to the number 432

1. Apr 14, 2013

fk378

If a,b are positive integers and (a1/2b1/3)6 = 432, then what is the value of ab?

2. Apr 14, 2013

Staff: Mentor

Is this a problem from the SAT?

First bring the 6 inside the a and b term to get a^6/2 * b^6/3 = 432

3. Apr 14, 2013

fk378

Yes, I did that. Don't know where to go from here. Seems like I just go in circles when I try to make two equations to solve for the two unknowns.

4. Apr 14, 2013

fk378

Is the only way to do this just to get a3b2=432, then find the factors of 432? I tried this and then got 16*27=432, so then a=3, b=2. But I feel like there must be a different way to do this problem...

5. Apr 14, 2013

L4xord

Hmmmm...... It would be good if there was another way. It seems a bit too easy.

6. Apr 14, 2013

Mentallic

You mean b=4

I'm not aware of another way if there is one, and I'd imagine if there were, it'd be fairly more complicated.

7. Apr 15, 2013

pwsnafu

We are given
$(a^{1/2} b^{1/3})^6 = 432$
So
$a^3 b^2 = a(ab)^2 = 432$
$(ab)^2 = \frac{432}{a}$
LHS is a square, so test different a.
$a = 2 \implies \frac{432}{a} = 216$ not a square
$a = 3 \implies \frac{432}{a} = 144$
144 is a square, so ab = 12.

8. Apr 15, 2013

Mentallic

Nice!

9. Apr 15, 2013

HallsofIvy

Staff Emeritus
Notice that the condition "a,b are positive integers" is crucial here. If a and b were allowed to be negative, there would be more solutions. If a and b were allowed to be any real numbers there would be an infinite number of solutions.

10. Apr 15, 2013

pwsnafu

If a and b were allowed to be negative and we are allowed to use complex algebra, then yes.
Otherwise a1/2 is undefined.