<p>I was taking one of the blue book's SAT tests when I came across this "hard" SAT math question. I am pretty good at math and usually get all the hard questions right, but this one blew me away. I know there is probably a very simple solution, but I cant seem to find it. Anyway, here it is. Thanks for helping!
BTW - PLease give me an explanation on how you got the answer. </p>
<p>If a and b are positive intergers and (a^1/2 * b^1/3) = 432, what is the value of ab?
A- 6
B-12
C-18
D-24
E-36</p>
<p>You miswrote the question: It should actually be 432^(1/6). That may be why you were having a hard time. If not, then I know how to solve it, and I'll gladly help. A quick hint: the exponent on the 432 should be your guide to solving the problem (i.e. try to get RID of it!!!). ;)</p>
<p>Oops... I did write the question wrong. The real questions is ....
If a and b are positive intergers and (a^1/2 * b^1/3)^6 = 432 what is the value of ab?
A- 6
B-12
C-18
D-24
E-36</p>
<p>Yoshi... I am kinda confused by your post. The question does not say 432^1/6. Are you looking in the Blue Book, test 4 (thats where I got it from). Sorry Yoshi .. still confused =(</p>
<p>Note in particular how 432 = 2^4 times 3^3, or:</p>
<p>432 = 2 * 2 * 2 * 2 * 3 * 3 * 3</p>
<p>Your number that is cubed (a) must be 3. But what about the number that is squared (b)? Simply group two pairs of twos together to find that number, which is 4:</p>
<p>432 = (2 * 2) * (2 * 2) * (3 * 3 * 3)</p>
<p>432 = (2 * 2)^2 * (3 * 3 * 3)</p>
<p>432 = (4)^2 * (3)^3</p>
<p>Therefore, a = 3 and b = 4, and ab = (3)(4), or 12.</p>
<p>Choice (B) is correct. Simplifying the exponential part of the expression gives (a^(6/2)<em>b^(6/3)) =a^3</em>b^2=432. The prime factorization of 432 is 3^3<em>2^4 so a^3</em>b^2=3^3<em>2^4. Since a & b must be positive integers, it follows that a^=3^3 and b^2=2^4=(2^2)^2 This yields a=3 and b=4 The question asks for the product of a and b which is 3</em>4=12</p>
<p>^Quix - it's unlikely you'll get an answer from Harry: his last post was on 09-17-2007.
I'd like to help you with this, but I don't quite understand your question.</p>
<p>===============
A while ago somebody posted a very efficient SAT-ish solution to this question, but somehow it never got into the Consolidated List.
Recently a student of mine came up with the same one, and I find it's worthwhile to post it here.
(a^1/2 * b^1/3)^6 = 432
a^3*b^2 = 432
(ab)^2 * b = 432
Let's examine given answers for ab:
A- 6
B-12
C-18
D-24
E-36
C is too big (18^2 = 324), so are D and E; A is too small; that leaves B.
If you are pressed for time you don't even need to find a and b.</p>
<p>Just in case:
ab = 12
12^2*a = 432
a = 432/144
a = 3
b = 4</p>