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When you have powers of 10, e.g. 102, the base is 10, so when the exponent is 2 you should not say power of 2. I believe "power of" refer to the base not to the exponent.

Mari-Lou A
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Maria
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    Whether it's "power" or "powers" depends on the context, not the base. – Hot Licks Mar 05 '19 at 12:52
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    Power in to the power of refers to the exponent as a location or role, which is also called exponent. This is only one for the operation denoted by 10^2: ten to the power of two. In powers of ten, the word powers refers to the collection of results obtained by raising ten to the different integer exponents. – user337391 Mar 05 '19 at 13:18
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    Every base is base 10. – Pieter B Mar 05 '19 at 15:50
  • What context do you need this in? If I understand you correctly there’s no concise idiom because there isn’t much reason (mathematically) to speak of “things raised to the nth power” as a unit, in the same way as we speak of “powers of n”: the mathematical series “powers of n” has important properties and connections to fundamental laws in mathematics. The series “things raised to the nth power” does not. – Konrad Rudolph Mar 05 '19 at 17:02
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    Possible duplicate of How to read exponential expressions, e.g., "2^16"? – JJJ Mar 05 '19 at 17:19
  • The example 10 exp 2 is not a good example, because the usual way to express that is 10 squared. 10 exp 3 is 10 cubed. Avogadro's Number is, on the other hand, 6.03 times 10 to the 23rd. As for every base being base 10, often the base is e (2.71828.....); sometimes the base is 2. The base can be anything, – ab2 Mar 05 '19 at 17:37
  • @PieterB: Ha! Good one. – Drew Mar 05 '19 at 18:54
  • @ab2 6.022 or round down, 6.02. Not 3. Huge difference when multiplying by 10^23. :) – JTP - Apologise to Monica Mar 06 '19 at 11:58
  • @JoeTaxpayer Thanks! My memory is failing me......:) – ab2 Mar 06 '19 at 13:38
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    @ab2 "Every base is base 10" was a joke. How do you write 2 in base 2? You write 10. – David K Mar 06 '19 at 13:54
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    "I believe ... ." When communicating, it's not very important what you believe the words you say mean; what really matters is what other people believe they mean. – David K Mar 06 '19 at 13:57
  • @David K Blush. – ab2 Mar 06 '19 at 22:38

8 Answers8

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I express 3^4 as “three to the fourth power”

You can say “base to the nth power” or “base to the power of n”

It’s important to have the whole sentence to determine if it makes mathematical sense.

JTP - Apologise to Monica
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20

While "ten to the power of two" is correct (and the "power" does indeed refer to the "two" in this construction), it's also possible and very common to drop the "power of", giving "ten to the two". When reading out vacuum pressures for example, "ten to the power of minus six" would never be heard from a native speaking physicist; we'd just say "ten to the minus six". This is equally true in longer constructions like "three point five times ten to the minus seven".

Chris H
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    It would generally be "to the minus sixth", or "10 to the 23rd". – jamesqf Mar 05 '19 at 19:20
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    @jamesqf not IME. I rarely hear positive ordinals used in this case for magnitudes, and never negative ordinals, unless followed by "power" (UK here). Also never "second", even "second power" - "squared" is occasional. When I say never I mean not once in my recollection in two physics departments and an engineering firm; I have a vague recollection that some of the older academics did use ordinals for magnitudes in the 90s. In formulae (x^5) cardinals are still more common but ordinals familiar - again only positive ones. – Chris H Mar 05 '19 at 19:49
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    @jamesqf I've never heard using ordinal numbers for negative powers: "ten to the third", but "ten to the minus three" (and I work in a field where powers of ten are fairly common). – Massimo Ortolano Mar 05 '19 at 22:33
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    @jamesqf "ten to the twenty-third power", or "ten to the twenty-three" - but not normally "ten to the twenty-third". So, "ten to the third" is 10^⅓, not 10^3 – Chronocidal Mar 05 '19 at 23:40
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    I disagree with @Chronocidal ... "ten to the third" means 10^3. – GEdgar Mar 06 '19 at 01:33
  • 10^⅓ would be "ten to the one third" if it was unavoidable (@Chronocidal) and 10^3 could just about be "ten to the third" but "ten to the three" is more common – Chris H Mar 06 '19 at 06:50
  • @Massimo Ortolano: I agree that sometimes it's cardinals with negative powers, but usually only for small powers. Seems to shift around -10. I don't recall ever hearing other than ordinals for positive exponents (other than "squared" or "cubed") - and I do have education/career in science. – jamesqf Mar 07 '19 at 00:47
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    @Chronocidal: No, 10^⅓ would be "ten to the one-third", at least as I would say it. 10^3 would most likely be "ten cubed", possibly "ten to the third", never "ten to the three". – jamesqf Mar 07 '19 at 00:50
  • @jamesqf where are you (if you don't mind me asking), because there seems to be a difference in actual use that may be regional - as I commented, cardinals seem to be the norm in my UK experience, with the exceptions being older – Chris H Mar 07 '19 at 06:45
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    @Chris H: Western US, mostly, with a couple years working with a research group in Switzerland. So indeed it might be a British/American difference. – jamesqf Mar 08 '19 at 04:11
12

The term power refers to the exponent, not to the base.

10 to the power 2 is 100.

However powers of 10 are the products obtained from raising 10 by various exponents. So again, power does not refer to the base.

Weather Vane
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    also power of 2 sometimes – gen-ℤ ready to perish Mar 05 '19 at 00:53
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    I always hear this phrased with ordinals rather than cardinals. – chrylis -cautiouslyoptimistic- Mar 05 '19 at 03:15
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    @ChaseRyanTaylor Power of 2 refers to binary arithmetic and means 2 raised to some power. Whenever we say "a power of n" we mean n raised to some power, that n is the base. The second power of 2 is 2x2 which is 4, the third power of 2 is 2x2x2 which is 8 and so on. Similarly the second power of 8 is 64. – BoldBen Mar 05 '19 at 09:35
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    @chrylis Both cardinals and ordinals are fine for integer and simple variable powers, "x to the sixth power", "x to the nth power" and "x to the power n" are all absolutely acceptable. However if you need to use negative, fractional or irrational powers then ordinals rapidly become clumsy. "n to the power minus two upon three" or "x to the power pi" are much clearer than "n to the minus two-third-th power" or "x to the pi-th power". – BoldBen Mar 05 '19 at 12:42
  • @chrylis that may be so but how would you say 10^0? It can be the same with percentages. Some people say "the 5th centile/percentile" but it gets tricky for a non-integer value such as 1.5 and in this case the more usual "1.5 percent" works better. The question however is about the use of the word power rather than whether cardinal or ordinal numbers are be used to express them. – Weather Vane Mar 05 '19 at 13:28
  • "Ten to the zeroth power." – chrylis -cautiouslyoptimistic- Mar 05 '19 at 13:30
5

A common expression for power(s) of 10 in regular speech is order(s) of magnitude.

From Wikipedia:

An order of magnitude is an approximate measure of the number of digits that a number has in the commonly-used base-ten number system. It is equal to the logarithm (base 10) rounded to a whole number. For example, the order of magnitude of 1500 is 3, because 1500 = 1.5 × 10^3.

Jase
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    be careful as some fields consider powers of 2 to be orders of magnitude. – james Mar 05 '19 at 09:28
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    @james could you give some examples of "powers of 2 = orders of magnitude" please? As a physicist (and formerly engineer) with a fair bit of software background and even some knowledge of bus-level data transport and machine code I've never come across this use. – Chris H Mar 05 '19 at 09:45
  • @ChrisH Well for one, SI units use orders of magnitude of base 1000, another is IEC prefixes which use orders of magnitude of base 1024 (admittedly can't think of any that use base 2 though). If I talk about an order of magnitude larger than MiB I don't think 10s of MiB, I think GiB –  Mar 05 '19 at 15:07
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    @NickA the second of those is possible, but the in usual use of "order of magnitude" 1000=1024 anyway (if I say my new hard drive is 3 orders of magnitude bigger than my old one, whether I'm referring to GiB and TiB or GB and TB is irrelevant). So 2^10 is 3 orders of magnitude, fine. But james didn't say that, instead implying something more like an order of magnitude means a doubling – Chris H Mar 05 '19 at 15:08
  • @ChrisH Yes sorry, I corrected myself on the GB/GiB, force of habit. I also see now you weren't arguing that others aren't valid, but base 2 specifically :), which I agree with. Although I would call 2^10 one order of magnitude rather than 3 (speaking in the context of where it makes sense) –  Mar 05 '19 at 15:08
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    @NickA further, the SI handbook, which I have in front of me, is perfectly happy with cm (it lists prefixes for 10^±1 and 10^±2 before the sequence of 10^3n). Ah OK, it sounds like we're not far apart. – Chris H Mar 05 '19 at 15:12
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    @ChrisH True, and I can't imagine (in terms of orders of magnitude) anything being more confusing than cm=-2, dm=-1, m=0, dam=1, hm=2, km=3, Mm=4..., So I retract the SI units one, although IEC applies :) (da and h I'd never heard of before now...) –  Mar 05 '19 at 15:15
  • Also, the magnitude of a star is the negative logarithm of its brightness (the idea is that the magnitude measures how big of a telescope you need to see it). – Acccumulation Mar 05 '19 at 20:21
4

Surprisingly, this is explained fairly well on Wikipedia.

I believe "power of" refer to the base not to the exponent

Nope. The spoken forms of 102 are:

  • 10 raised to the second power, or
  • 10 raised to the power of two, or
  • 10 to the power of two, or
  • 10 to the two, or simply
  • 10 squared

Since the original formulation base raised to the nth power means multiply 1 by base n times, the word power does indeed refer to the exponent.

Useless
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  • "Raised" and "of" are optional in your examples. E.g., "x to the power y" is completely normal in mathematical English. – David Richerby Mar 05 '19 at 17:01
  • I think I'd usually omit the "power" as well in that case, but pretty much any combination seems acceptable. – Useless Mar 05 '19 at 17:17
4

"Powers of 10" does definitely refer to power expressions with 10 as a base rather than as an exponent. I don't have any sourced explanation (which makes this a terrible answer), but I imagine it's because of the similarity between the two phrases

  • 10 raised to the second power
  • the second power of 10
Arthur
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2

The expression a power of 10 typically means the number you get when you raise 10 to a power (exponent, in other words) which itself is a number. I know it's a little bit confusing since you refer to the result of raising a number to a power also as a power, but that's just how people say it. Thus, you can say that the following is a list of powers of 10, that is, a list of the numbers you get when you raise 10 to a particular power such as 1, 2, 3, etc:

101 = 10
102 = 100
103 = 1000
etc.

Given the fact that the numbers 102 and 100 are equivalent, they both can be referred to as a power of ten. More specifically, it's ten raised to the second power or more compactly ten to the second power. Likewise, 108 would be pronounced ten to the eighth power or ten raised to the eighth power.

Usually, for powers that are greater than 3, you can drop the word "power". For example, instead of saying ten to the eighth power, you can just say ten to the eighth.

Michael Rybkin
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1

Colloquially, 10^2 is often called "10 to the power of 2"; but as others have noted, the "öf"is redundant, and strictly incorrect. As you suggest, it is better to keep the phrase "power of" to refer to the base; and say "10 to the power 2", or just "10 to the 2".

Keith Anker
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