# How is the word "power" used in mathematics?

When I was researching on the Web, I came across the following expression in a sentence. (Because the entire sentence was technical, I didn't mention the entire sentence)

"a base expression taken to a specified power."

Then I got curious to know how the word "power" is used in English (mathematics). You know, in math, when you take a number to a specific power, it is multiplied by itself and the number of times that it is multiplied by itself depends on the exponent.

Now I want to know more about how the word "power" is used in English. For example, how do you say a mathematical expression in which power is used? Or how do you ask somebody to tell the result of a number taken to a specific power?

The term that you are looking for, the result when a base is "raised to a power" is power … and it is incorrect to say "a base raised to a power". We should say "a base raised to an exponent, which is a power". The term power is one of the most widely misused in all of mathematics. Let me explain.

Let’s say that Tom has 5 children. They are, in order starting the the first: A, B, C, D and E. The third child of Tom is C. Tom is the parent of this list or sequence of children. We will then refer to Tom as the "root" or the "base".

Next, the multiples of 2 are: 2, 4, 6, 8, 10, … This sequence is based on 2 (it starts at 2 and we add 2 to get each successive number), so 2 is the base. The third multiple of 2 is 6. 2 + 2 + 2 = 6. We can condense this as 3x2 = 6 (3 copies of 2, added together equals 6). 6 is the sum, or in the case of repeated addition, the multiple. Of whom? Of 2. Which multiple? The third one.

The powers of 2 are: 2, 4, 8, 16, 32, … The base of this sequence is 2 because we start at 2 and multiply by 2 to get each successive number. The third power of 2 is 8. 2 x 2 x 2 = 8. We can condense this as 2^3 = 8 (3 copies of 2, multiplied together equals 8). 8 is the product, or in the case of repeated multiplication of a base number, the power. Of whom? Of 2. Which power? The third one.

Let’s recap;

1. C is the third child of Tom.
2. 6 is the third multiple of 2.
3. 8 is the third power of 2.

So who is the third

1. child? C … not 3
2. multiple? 6 … not 3
3. power? 8 … not 3! 3 is the *not the power, 3 is the exponent! 8 is the power!

In 2^3 = 8, 2 is the base, 3 is the exponent and 8 is the power. Specifically 8 is the third power of 2.

The term power is the most widespread and consistently grammatically misused word in all of mathematics. The power is NOT the exponent, ever. You cannot read 2^3 as "2 raised to the power 3" or "2 raised to the third power" or "2 to the third" (that sounds like 2^(1/3) which is the cube root of 2, not 2^3 = 8).

The third power of 2 is 8. So 8 is the power when 2 is the base and 3 is the exponent.

Read 2^3 as "2 raised to the exponent of 3", "2 to the 3" (though I prefer that learners emphasise that 3 is the exponent) or, in this specific example, "2 cubed".

It may seem like splitting hairs to distinguish between "exponent" and "power" but it is crucial to understanding exponential and logarithmic functions.

If y = 2^x, then for the exponential base 2 function, the input x is an exponent and the output is a power. If we cannot distinguish inputs from outputs, deep and authentic understanding cannot be achieved.

It would be like saying in 3x5=15 that 15 is the product. And so are 3 and 5. Which is false, because product is the output of multiplying two (or more) input factors together. It would be like saying that area of a rectangle = length x width: A = l x w, so the output A is area … and so are the inputs l and w also area!

So, b^x = p says that b is the base, x is the exponent and p is the power … regardless of what you read on the internet.

You know, in math, when you take a number to a specific power, it is multiplied by itself and the number of times that it is multiplied by itself depends on the exponent.

That's basically all there is to it.

If you have the mathematical expression 23, you can say it aloud as either

two to the power of three

or

two to the third power

If you working in a field where you use math a lot, you'll probably most often hear the second version shortened to just "two to the third".

• "Two to the power of three" Thank you for your answer. Somebody also sent me an answer and told me that I can say "two to the power three" (he or she didn't put "of" between "power" and "three"). Is it incorrect? By the way, his or her answer was deleted immediately! I don't know why. Jan 2, 2018 at 10:28

"Power" may be used in at least two senses in mathematical English. For example, there are "power sets" in set theory. Mathematics is filled with definitions, and two sub-fields may use the same word in different senses. However, the value of an exponent is far and away the most common mathematical meaning of "power." One might go so far as to say it is the sole meaning in elementary algebra and calculus.

• Ok, thank you, I read your answer. So the word "power" has at least two meanings in mathematics. But can you explain what are "power sets" in set theory? Jan 2, 2018 at 11:12
• Here is a citation to power sets: en.m.wikipedia.org/wiki/Power_set I want to stress that the primary mathematical meaning of "power" is the value of an exponent. But different branches of mathematics can use the same word with different meanings. Jan 2, 2018 at 14:08