<p>from pg 714 in blue book</p>
<ol>
<li>if a and b are positive integers and ((a^1/2)(b^1/3))^6 = 432, what is the value of ab</li>
</ol>
<p>A 6
B 12
C 18
D 24
E 36</p>
<p>solve for a and substitute</p>
<p>Since the whole thing is to the sixth do “power to the power” so a^3<em>b^2=432. Now you may realize that a could =3 and b could =4. So 3^3 * 4 ^2 = 432! So ab=3</em>4=12. The correct answer is B!</p>
<p>Simplify the left side</p>
<p>a^3 * b^2 = 432</p>
<p>The left side looks like a factorization, so try writing out the prime factorization of 432</p>
<p>432 = 216 * 2 = 108 * 2 = 54 * 2 = 3^3 * 2 hence</p>
<p>432 = 3^3 * 2^4
2^4 is the same as 4^2 so
3^3 * 4^2 = 432, in the same form of
a^3 * b^2 = 432</p>
<p>in which case a = 3, b = 4 and ab = 12.</p>
<p>Okay one question. How do you know that 3^3*4^2 equals 432? Trial and error? intuition?</p>
<p>I found the prime factorization of 432 and went from there…</p>