<p>I'm looking at the blue book page 714, prob.8. I can get the answer by just putting different numbers in. But is there a better way to do it?</p>
<p>Virtually every math problem can be solved without plugging in a bunch of numbers…if you could post the problem here I’d be happy to help (sorry I don’t have a blue book).</p>
<p>Here is the problem statement:
If a and b are positive integers and ([a^1/2][b^1/3])^6=432, what is the value of ab?</p>
<p>First simplify the exponent expression to: (a^3)(b^2)=432, and because “a” and “b” are integers we know that doing the prime factorization of 432 should help. 432 =(6)(72)=(2)(3)(9)(8)=(2^4)(3^3)</p>
<p>By observation we see that a=3 and b=4=2^2, meaning (3^3)(4^2)=(a^3)(b^2)=432 and ab=12. </p>
<p>Another approach is to somehow create the term ab in the given equation:
(a^3)(b^2)=432 </p>
<p>Divide both sides by “a”</p>
<p>(a^2)(b^2) = 432/a = [(2^4)(3^3)]/a</p>
<p>The term [(2^4)(3^3)]/a has to be a perfect square, which means “a” must be equal to 3, and therefore (a^2)(b^2) = [(2^4)(3^2)] or ab=(2^2)(3)=12.</p>
<p>Starting from (a^3)(b^2) = 432 = (2^4)(3^3), notice that the only cubes that are factors of 432 are 1, 3^3, and 6^3. However a = 6 doesn’t work, as this implies b^2 = 2. Also, a = 1 doesn’t work, since b^2 = 432. Now we know that the <em>only</em> possibility for a is 3, so b = 4 and ab = 12.</p>
<p>Thanks guys. These are very helpful.</p>