Power Conversation

[Steven G]

Here is the property that deals with negative exponents:

x^(-n) = 1/x^n

In the expression 6*x^(-2), the power is x^(-2). The exponent does not apply to the 6 because the Order of Operations tells us to do exponentiation before multiplication.

If we had (6x)^(-2), instead, then the exponent applies to both the 6 and x.

(6x)^(-2) = 1/(36x^2)

Cheers :cool:

Otis, no the power is NOT x^(-2) as the problem is not 6^(x^(-2)). The power for x is -2, ok?

 

[Otis]

The power for x is -2, ok?

Hi Jomo. I'm not sure whether I'm okay with that.

Such a view seems to require the words 'power' and 'exponent' to be synonomous. I learned a different interpretation.

In my view, a power is a repeated multiplication, like 4*4*4*4*4.

The shorthand notation for a power requires two parts -- a base and an exponent.

4*4*4*4*4 = 4^5

The blue part is the base, and the red part is the exponent.

I've never seen the red part referred to as a power, but that doesn't say that some folks don't.

I have heard statements like "4 raised to the power of 5", but do you see the difference? I don't read that phrase to say that 5 is the power of 4.

The distinction that I desired to make regarding the expression 6*x^(-2) is that it's comprised of two separate parts -- a coefficient and an exponentiation. The exponentiation does not affect the coefficient (they stand apart). In other words, I want students to be mindful of the Order of Operations.

I always hope that my wording does not confuse students; I will reflect on your point further. Thanks.

 

[Steven G]

I have heard statements like "4 raised to the power of 5", but do you see the difference? I don't read that phrase to say that 5 is the power of 4.

4 to the 5th power, 4 raised to the 5th power, 4 to the power of 5,... all say that 5 is the power. Possibly in different parts of the world they say it your way. I did not know that and have absolutely no problem if that is the case. Well, actually one problem and that is it would be better if the mathematical world would be consistent so we can communicate as one.

 

[Otis]

4 to the 5th power, 4 raised to the 5th power, 4 to the power of 5,... all say that 5 is the power.

Interesting, to be sure, Jomo. In my view, these are all different versions of saying that a power exists (five factors of four multiplied together), and the notation 4^5 immediately comes to mind. Yet, I do not consider five to equal the power described by these statements. Five is a required part of the description.

Let's look from the other direction. Please consider the following statement.

"5 is an exponent."

Now, in the absense of any additional information, would you say that this 5 itself is a power, or would you say that it's a power OF something?

Or, consider the following exercise.

"What does the power 4^5 equal?"

You would not answer 5, right? (Perhaps, you would claim that the exercise is nonsensical.)

The distinctions that we're each trying make may boil down to a matter of accepted semantics. Like interchanging the words 'percent' and 'percentage', as though they have the same meaning. (To me, a percentage is a percent OF something, not the percent itself, which is a conversion factor; yet, others seem to get by using these as two different spellings of the same word). Or, like saying that 4^5 means four multiplied by itself five times (just plain wrong, in my view, but makes sense to a lot of people nonetheless because the wording is close enough to form a correct picture of 4*4*4*4*4).

I hope that my wording about 6*x^(-2) is close enough, too. The picture that I want to communicate is that 6 is not part of the base, in that exponentiation.

it would be better if the mathematical world would be consistent so we can communicate as one.

I don't see this ever happening. Once humans are eliminated, however, the machines will be able to communicate as one.

PS: Where is your part of the world? I'm in North America. Cheers!

 

[Steven G]

Interesting, to be sure, Jomo. In my view, these are all different versions of saying that a power exists (five factors of four multiplied together), and the notation 4^5 immediately comes to mind. Yet, I do not consider five to equal the power described by these statements. Five is a required part of the description.

Let's look from the other direction. Please consider the following statement.

"5 is an exponent."

Now, in the absense of any additional information, would you say that this 5 itself is a power, or would you say that it's a power OF something?

Or, consider the following exercise.

"What does the power 4^5 equal?"

You would not answer 5, right? (Perhaps, you would claim that the exercise is nonsensical.)

The distinctions that we're each trying make may boil down to a matter of accepted semantics. Like interchanging the words 'percent' and 'percentage', as though they have the same meaning. (To me, a percentage is a percent OF something, not the percent itself, which is a conversion factor; yet, others seem to get by using these as two different spellings of the same word). Or, like saying that 4^5 means four multiplied by itself five times (just plain wrong, in my view, but makes sense to a lot of people nonetheless because the wording is close enough to form a correct picture of 4*4*4*4*4).

I hope that my wording about 6*x^(-2) is close enough, too. The picture that I want to communicate is that 6 is not part of the base, in that exponentiation.




I don't see this ever happening. Once humans are eliminated, however, the machines will be able to communicate as one.

PS: Where is your part of the world? I'm in North America. Cheers!

Otis, I live in New York State.
(oh my, my post count is 666)

 

[Otis]

my post count is 666

That's my current credit score :lol: