Finding the value of...

**Hellbent** · December 30th 2010, 01:10 AM

Hi,

$\mathrm{If\ a\ and\ b\ are\ positive\ integers\ and\ (a^\frac12 \cdot b^\frac13)^6\ =\ 432\ what\ is\ the\ value\ of\ ab?}$

$(a^\frac12 \cdot b^\frac13)^6\ =\ 432$

$\underbrace{a^3\ \cdot\ b^2\ =\ 432}_\mathrm{Stuck\ here}$

Perhaps I went about it incorrectly...

**TheCoffeeMachine** · December 30th 2010, 01:23 AM

$432 = 2(216) = (2)(2)(108) = 2(2)(2)(54) = 2(2)(2)(2)(27) = (4)^2(3)^3.$

**Hellbent** · December 30th 2010, 02:11 AM

So ab = 12. I figured this out by removing the exponents from both sides, knowing that the bases on both sides would become equal; seeing that this is math, magic isn't allowed.

So I ask, what are the intermediary steps between:
$a^3 \cdot b^2 = (4)^2(3)^3$ that math allows?

**TheCoffeeMachine** · December 30th 2010, 01:14 PM

Originally Posted by Hellbent

So I ask, what are the intermediary steps between:
$a^3 \cdot b^2 = (4)^2(3)^3$ that math allows?

What you did is correct! Magic isn't allowed, but intuition is. We compare the bases: if $(a)^3(b)^2 = (4)^2(3)^3$ ,
then $a = 3$ , and $b = 4$ . It's purely deductive. (It's the same way we compare the coefficients of polynomials).

**Archie Meade** · January 1st 2011, 04:08 PM

Originally Posted by Hellbent

Hi,

$\mathrm{If\ a\ and\ b\ are\ positive\ integers\ and\ (a^\frac12 \cdot b^\frac13)^6\ =\ 432\ what\ is\ the\ value\ of\ ab?}$

$(a^\frac12 \cdot b^\frac13)^6\ =\ 432$

$\underbrace{a^3\ \cdot\ b^2\ =\ 432}_\mathrm{Stuck\ here}$

Perhaps I went about it incorrectly...

Since $a$ and $b$ are positive integers,

$a^3b^2=432\Rightarrow\ a^3b^3=432b\Rightarrow\ (ab)^3=432b$

which is a multiple of 432 and $ab$ is also an integer.

Multiples of 432 are $432(2)=864,\;\;432(3)=1296,\;\;432(4)=1728,....$

$7^3=343,\;\;8^3=512\ \ne\ 432k$

$9^3=729,\;\;10^3=1000,\;\;11^3=1331,\;\;12^3=1728$

We have a hit.

$ab=12$

Alternatively,

$\displaystyle\ a^2b^2=\frac{432}{a}\Rightarrow\ (ab)^2=\frac{432}{a}$

$\displaystyle\frac{432}{2}=216,\;\;\frac{432}{3}=1 44,......$

$12^2=144\Rightarrow\ ab=12$

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