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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.

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    Whether it's "power" or "powers" depends on the context, not the base. – Hot Licks Mar 5 '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 5 '19 at 13:18
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    Every base is base 10. – Pieter B Mar 5 '19 at 15:50
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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 6 '19 at 13:54
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".

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    It would generally be "to the minus sixth", or "10 to the 23rd". – jamesqf Mar 5 '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 5 '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 5 '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 5 '19 at 23:40
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    I disagree with @Chronocidal ... "ten to the third" means 10^3. – GEdgar Mar 6 '19 at 1:33
20

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.

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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.

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    also power of 2 sometimes – gen-z ready to perish Mar 5 '19 at 0:53
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    I always hear this phrased with ordinals rather than cardinals. – chrylis -on strike- Mar 5 '19 at 3:15
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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 5 '19 at 9:35
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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 5 '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 5 '19 at 13:28
4

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.

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    be careful as some fields consider powers of 2 to be orders of magnitude. – james Mar 5 '19 at 9:28
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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 5 '19 at 9:45
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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 5 '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 5 '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...) – user300397 Mar 5 '19 at 15:15
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.

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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 5 '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 5 '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
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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.

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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".

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