# Exponent rules: what am I doing wrong? **URL:** \ **Category:** Factual Questions\ **Tags:** science-math\ **Created:** [April 7, 2025, 3:54pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742 "2025-04-07T15:54:08Z")\ **Posts on this page:** 17\ **Page:** 1
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [April 7, 2025, 3:54pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/1 "2025-04-07T15:54:08Z")
(Urgh, I am not getting markup to work, so apologies for the clumsy notation) So one of the exponent rules is for powers of a power: A^(b)^(c) = A^(b×c). Clearly this works; for example 3^9 equals 3×3×3×3×3×3×3×3×3 which equals 3^(3)^(3) = 3^(3×3). Fine. My problem is that 9 indisputably equals 3 squared: 3^2. But by that reasoning 3^9 should equal 3^(3)^(2), which is flatly in contradiction to the preceding. I **know** that’s wrong; but I can’t explain why it’s wrong. Why is it now forbidden to say that nine equals three squared just because the nine is itself an exponent? ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [April 7, 2025, 4:01pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/2 "2025-04-07T16:01:51Z")
Order of operations matters. Exponentiation is evaluated from right to left. So, 3^3^3 is interpreted as 3^(3^3), which is 3^27 (about 10^13). This is obviously different than (3^3)^3, which is 27^3 (about 10^4). For clarity, it’s good to always include parentheses or other notation to make explicit the order of operation. ---
**Author:** ![Jasmine](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jasmine/32/2964_2.png) [@Jasmine](https://boards.straightdope.com/u/Jasmine)\ **Post date:** [April 7, 2025, 4:02pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/3 "2025-04-07T16:02:01Z")
**When multiplying like bases, keep the base the same and add the exponents**. When raising a base with a power to another power, keep the base the same and multiply the exponents. When dividing like bases, keep the base the same and subtract the denominator exponent from the numerator exponent. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [April 7, 2025, 4:40pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/4 "2025-04-07T16:40:53Z")
> [@Pleonast](#): > > Exponentiation is evaluated from right to left. In computer languages, where ambiguity is a Bad Thing, this is the standard rule for the exp operator. The other common operators are the other way around. E.g. 48/6/2 is 4 not 16. ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [April 7, 2025, 4:59pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/5 "2025-04-07T16:59:30Z")
So far so good, but here I was saying to myself “okay, I have 3^9, and 9 is 3^2 so I can–” **BBZZZZZTT!** WRONG! What happened? ---
**Author:** ![Topologist](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/topologist/32/3208_2.png) [@Topologist](https://boards.straightdope.com/u/Topologist)\ **Post date:** [April 7, 2025, 5:04pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/6 "2025-04-07T17:04:36Z")
The markup may make it clearer: Yes, 3^9 = 3^{3^2}, or 3^(3^2) not using the markup. But the parentheses are important. That’s different from (3^3)^2, or (3^3)^2, which is the same as 3^6. If you simply write 3^3^2 (which the markup won’t accept), it’s ambiguous until you accept a convention on whether to read it left-to-right or right-to-left. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [April 7, 2025, 5:05pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/7 "2025-04-07T17:05:14Z")
> [@Lumpy](#): > > But by that reasoning 3^9 should equal 3^(3)^(2) Nope. You reasoning leads to 3^(3^2) as in do the squaring of the 3 first. ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [April 7, 2025, 5:12pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/8 "2025-04-07T17:12:40Z")
Okay, so what I’m hearing is that without an agreed order of operations or parentheses, stacked exponents are ambiguous. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [April 7, 2025, 5:16pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/9 "2025-04-07T17:16:29Z")
Except it is agreed to. Exponents are right-associative meaning 2^3^4^5^6 would start with the 5^6, then 4^15625 &c. ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [April 7, 2025, 5:29pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/10 "2025-04-07T17:29:49Z")
This order of operations difference surprises me because one would think that exponentiation, being an increasing operation like addition and multiplication, would be commutative like they are. ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [April 7, 2025, 5:37pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/11 "2025-04-07T17:37:34Z")
The commutative property is a separate issue, meaning A _op_ B = B _op_ A for some operator _op_. For order of operations, exponentiation is naturally right to left with the common notation: 4^{3^{2^1}}. You can’t raise 4 to a power until you know what that power is. You can’t raise 3 to a power until you know what that power is. So you have to start with 2^1 = 2. Then 3^{2^1} = 3^2 = 9. Then 4^{3^{2^1}} = 4^9 = 262144. ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [April 7, 2025, 5:39pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/12 "2025-04-07T17:39:56Z")
@Pleonast , that explains it very well, thank you. I strongly suspect that I would not do as well on a remedial algebra extension course as I had thought. ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [April 7, 2025, 5:44pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/13 "2025-04-07T17:44:30Z")
> [@Lumpy](#): > > I strongly suspect that I would not do as well on a remedial algebra extension course as I had thought. Mathematical notation is just a convention for expressing ideas precisely. It’s not actually math. So, you might do the math well, once you get past the initial barrier of how to write stuff down. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [April 7, 2025, 6:49pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/14 "2025-04-07T18:49:49Z")
Obligatory mention that we should go to RPN to avoid all of this nonsense. Not that I believe that but we all know it’s going to show up eventually so we might as well get it out of the way early. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [April 7, 2025, 7:16pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/15 "2025-04-07T19:16:24Z")
The issue here is that it’s not associative. (It’s not commutative either. ) ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [April 7, 2025, 7:19pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/16 "2025-04-07T19:19:23Z")
I’ll note that would be possible and consistent to make exponentiation be left-associative instead, but it would be less useful. For example, if exponentiation were left-associative, 4^{3^2} would be equal to ({4^3})^2, which is equal to 4^{3 \cdot 2}. It’s just as simple, or simpler, to write it as a single exponentiation of a product instead of a double exponentiation. So in that world, you would need to add parentheses to force right-associativity: 4^{(3^2)}, while in our world, parentheses are not needed to force left-associativity, you just write the exponent as a product. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [April 7, 2025, 9:17pm UTC](https://boards.straightdope.com/t/exponent-rules-what-am-i-doing-wrong/1016742/17 "2025-04-07T21:17:36Z")
> [@markn\_1](#): > > I’ll note that would be possible and consistent to make exponentiation be left-associative instead, but it would be less useful. Which gets to the point that the conventions of mathematical notation (all of them, not just this case) are not handed down from on high on sacrosanct stone tablets. Humans chose to set up those conventions, and they chose to set them up the way they did for the sake of convenience.