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(I see an error I made, but I'd still like to know if there is a specific order.)

I have here $\left(6^{-36}\right)/\left(6^{-16}\right)\cdot\left(6^{16}\right)$.

If I do the division first, it's $-36$ minus $-16$, making an addition of plus $16$ for $6^{-20}$.

This times the $6^{16}$ equals $6^{-4}$.

But if I do the multiplication first, the powers of $16$ and $-16$ cancel out, leaving me with $6^{-36}$, which is a radically different answer.

When it comes to exponents, do I have to strictly go from left to right first, in terms of division/multiplication?

user729424
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Dialectics
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1 Answers1

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Evaluating $a/b\cdot c$ requires a bracketing convention. On BODMAS we divide first, giving $(a/b)c=ac/b$. On PEMDAS we multiply first, giving $a/(bc)$.

J.G.
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  • Both of them are supposed to be equal though no? Both division and multiplication have the same priority, so basically im just wondering if we do left to right as convention? Meaning both bodmas and pedmas would be the same. As in, bodmas isnt always division first and pedmas isnt always multiplication first. – Dialectics Apr 01 '20 at 13:09
  • @Dialectics The end result is that it differs based on context and the notation is ambiguous. One should use enough parentheses so that what is written cannot be confused (while also not using so many that it becomes unreadable). All that is important for you right now is understanding that this is a problem with ambiguity so that you can avoid writing something like this, and understanding how to evaluate either interpretation, be it $a/(bc)$ or $(a/b)c$ – JMoravitz Apr 01 '20 at 13:14
  • @Dialectics No, they're not of the same priority: that's why the results are so different. – J.G. Apr 01 '20 at 13:26
  • @JMoravitz I was wondering how to tag someone haha. I understand how to evaluate a/(bc) and (a/b)c because the parenthesis make it obvious. Im not sure if its me who is going wrong or if the question in the textbook is simply too ambiguous. Im provided (6^9)^-4 / (6^2)^-8 * (6^4)^4 in the book. Literally how its written. When we apply the exponent laws, we get what i provided in OP. Either the brackets are written around the base and new powers or they are removed altogether. There are no brackets to indicate the order of division or multiplication. In a case like, is it right to left – Dialectics Apr 01 '20 at 13:30
  • @J.G. So essentially the question is bunk/too ambiguous? its asking me to reduce to a single power then evaluate. – Dialectics Apr 01 '20 at 13:32
  • As already said, in a case like this it is ambiguous. Know how to do it if you choose to interpret it from left to right. Know how to do it if you choose to interpret it from right to left. Know that the book did a poor job in writing the question by using ambiguous notation and that neither answer is "more correct" than the other and any answer given needs this warning that you assumed one or the other. Know that in the future when presented with a similar question similarly ambiguously worded that you will have the same problem. – JMoravitz Apr 01 '20 at 13:33
  • @JMoravitz Alright thank you man XD – Dialectics Apr 01 '20 at 13:34
  • There are some contexts where one reading is preferred or intended and the ambiguous notation is used despite the ambiguity. E.g., in the case of $360/2\pi$ it should be clear from context that we are talking about converting an angle from radians to degrees and that it is meant to be read $360/(2\pi)$. That doesn't necessarily excuse it as being poorly written. In the real world, if you ever see an expression written this way you generally know what the expression was meant to represent as well and so could understand the author's intention. – JMoravitz Apr 01 '20 at 13:38