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I was helping a high school student with their homework and I ran across the following problem: solve $$3^x+3^{(3x+1)}=108.$$ I was unable to find any "elementary way" to do this (by which I mean something a high school student would be comfortable with like manipulating the equation until the bases are the same and then equating the power, or using logarithms to undo exponents).

Can anyone solve this with $9^{th}$ or $10^{th}$ grade level math?

recmath
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    Replace $y=3^x$. You don't get a nice cubic, but it boils down to solving $y+3y^3=108$. Are you sure that this is the question? Cubics are always solvable by different methods, but in this case solutions are not pretty – chubakueno Aug 08 '14 at 00:18
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    Are you sure the question wasn't $3^x + 3^{x+1} = 108$ ? – DanielV Aug 08 '14 at 00:24
  • That is the method i used (i basically walked the student through cardano's formula). But the student lost interest and the teacher explicitly stated "show all steps" and no calculator. I too was thinking there was a typo but that was not comforting to the student. – recmath Aug 08 '14 at 00:24
  • I am sure about the question – recmath Aug 08 '14 at 00:25
  • There is nothing in the general process for solving a cubic equation that a 9th or 10th grade student couldn't understand. It's just an algorithm that only makes use of basic arithmetic and algebra. So following @chubakueno's comment, you could apply that method. The catch is that most 9th and 10th graders have not been taught that algorithm, so it's up to you to decide if that counts. – 2'5 9'2 Aug 08 '14 at 00:25
  • $3^x+3^{(2x+1)}=108$ is another possibility for a typo. – 2'5 9'2 Aug 08 '14 at 00:26
  • @alex.jordan unfortunately the student didn't care for my explanation of the cubic formula.. Furthermore the other questions on the homework were simpler and easily solved which makes me think the teacher made a typo .... However, the student was adamant that the problem was correct. – recmath Aug 08 '14 at 00:27
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    Definitely a typo! Or they want the student to use a calculator to solve... – ClassicStyle Aug 08 '14 at 00:35
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    @TylerHG certainly no calculators allowed.. And I'm glad others agree me..I will show this to the student and hopefully he will believe me! – recmath Aug 08 '14 at 00:36
  • There is no way in hell the student was expected to solve that cubic. That seems more of an annoying computation for undergrad algebra. From my own memories of typos and miscopies, I would bet 10 bucks that the problem should read either $3^{3x}+3^{3x+1} = 108$ or $3^{x}+3^{x+1} = 108$, with trivial solutions by observation 1 or 3, respectively, (or your substitution method to really clinch it, illysial), but still.... first/second year of average-track high school does not expect much more from exponentials. – Badam Baplan Aug 08 '14 at 00:42

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Follow chubakueno's suggestion of letting $y=3^x$, giving us

$$3y^3+y-108=0$$

The only hope to factor this is for the equation to have a rational root. According to the rational root theorem, such a root would have to be of the form $\frac pq$ such that $p$ is a factor of $108$ and $q$ is a factor of $3$. It's also relatively easy to see that there should be one real root between $3$ and $4$. Since our root is not an integer, we must test $q=3$. Since neither $10$ nor $11$ are factors of $108$, this root cannot be rational. High school algebra won't be able to solve this one.

Mike
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  • You need to assume $,p,q,$ coprime for RRT to apply, i.e. that $,p/q,$ is in lowest terms. – Bill Dubuque Aug 08 '14 at 00:58
  • @BillDubuque If $p$ and $q$ aren't coprime, the numerator and denominator of the reduced fraction should still be factors of the proper coefficients. – Mike Aug 08 '14 at 01:09
  • RRT requires coprimality, e.g. $,2/6,$ is a root of $,3x!-!1,$ but $,6\nmid 3,\ 2\nmid 1.\ $ Yes, wlog you may assume the fraction is in lowest terms. But you must say that to apply RRT. – Bill Dubuque Aug 08 '14 at 01:16
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    @BillDubuque I think you're just nitpicking. Is it so wrong that I find the root $1/3$ and miss the root $2/6$? They're equal. If you try all such $p$ and $q$ as laid out here, you will find all roots. It may be possible to write them in different forms, but who cares? The point is to set up a finite search space to find all roots. – Mike Aug 08 '14 at 01:46
  • This oversight can lead to subtle, serious errors in less trivial contexts. That's why when I teach RRT I always emphasize proper application. I recall one student who wasted a few days searching for a subtle error in his thesis that essentially boiled down to this oversight. But I can certainly understand why it may seem like nitpicking. – Bill Dubuque Aug 08 '14 at 01:59