Raising a power to a negative power

[Beowulf]

If I have the question what is 7^3^-2 and I do
7^3 = 343
343^-2 = .0000084

If I try to answer that by using the rule for raising a power to a power then I get:
7^3^-2
= 7^(3*-2)
= 7^-6
= 1/7^6 = .0000084

So I am pretty sure that I understand this correctly and yet if I go to any number of calculators, online, on my phone and pysical and enter
7^3^-2 I get the really confusing answer of 1.241365817.

If I enter it into the calculator as (7^3)^-2 then I get .0000084

So what is the difference between (7^3)^-2 and 7^3^-2?

 

[Deleted member 4993]

If I have the question what is 7^3^-2 and I do
7^3 = 343
343^-2 = .0000084

If I try to answer that by using the rule for raising a power to a power then I get:
7^3^-2
= 7^(3*-2) ......This is not correct

7^3^-2

= 7^ (3^-2)

= 7 ^ (1/9)

and continue.....


= 7^-6
= 1/7^6 = .0000084

So I am pretty sure that I understand this correctly and yet if I go to any number of calculators, online, on my phone and pysical and enter
7^3^-2 I get the really confusing answer of 1.241365817.

If I enter it into the calculator as (7^3)^-2 then I get .0000084

So what is the difference between (7^3)^-2 and 7^3^-2?

.

 

[stapel]

So what is the difference between (7^3)^-2 and 7^3^-2?

The former means this:

. . . . .\(\displaystyle (7^3)^{-2}\, =\, 7^{3\cdot-2}\, =\, 7^{-6}\, =\, \dfrac{1}{7^6}\)

The latter means this:

. . . . .\(\displaystyle 7^{3^{-2}}\, =\, 7^{\frac{1}{9}}\, =\, \sqrt[9]{\strut 7\,}\)

 

[Beowulf]

The former means this:

. . . . .\(\displaystyle (7^3)^{-2}\, =\, 7^{3\cdot-2}\, =\, 7^{-6}\, =\, \dfrac{1}{7^6}\)

The latter means this:

. . . . .\(\displaystyle 7^{3^{-2}}\, =\, 7^{\frac{1}{9}}\, =\, \sqrt[9]{\strut 7\,}\)

OHHHHH!

I was just coming up on that as I worked through the differences in the last person's reply. So if I raise x to a in brackets then raise the *result* by another exponent I would multiply. But if I raise x to a to b, then order of operations rules the day (as always) and I resolve a to the power of B first, then apply that result to x.

If that makes sense.

You have no idea how much that clears up and solidifies in my head.

I may be a long way from taking on calculus and physics, but I'll not let pre-algebra be where I fail to achieve complete mastery.

EDIT: Well, I guess order of operations always ruled the day. I guess this error was more along the lines of ignoring order of operations and trying to apply the ... distributive property?... where it didn't actually fit.

 

[mmm4444bot]

I agree with stapel, that (7^3)^(-2) means this:

. . . . .\(\displaystyle (7^3)^{-2}\, =\, 7^{3\cdot-2}\, =\, 7^{-6}\, =\, \dfrac{1}{7^6}\)

7^3^-2 means nested exponentiation, so we work from right to left.

Stapel is correct; without grouping symbols, it's the same as 7^(3^(-2))

(Note: I have edited this post because I initially got the order backwards. Doh!)

(I

 

[Beowulf]

That was my initial reasoning, but two calculators (a TI and a Casio), and wolframalpha all calculate 7^3^(-2) as the 9th root of 7.

So I believe the current convention is that if there are no parenthesis you start with the 'highest' exponent and compute down.

 

[mmm4444bot]

That was my initial reasoning, but two calculators (a TI and a Casio), and wolframalpha all calculate 7^3^(-2) as the 9th root of 7.

Oh, of course. I was not thinking clearly; I apologize for any confusion, and I will edit that post.

These are like nested exponentials, so we need to work from the "inside out". Without grouping symbols, that means from right to left!

So I believe the current convention is that if there are no parenthesis you start with the 'highest' exponent and compute down.

The highest exponent is 3. If we start with that, we get 343^(-2). That's not the 9th root of 7. It's the reciprocal of 7^6.

The convention is to work from right to left.

Better convention: use grouping symbols or fancy mathematical formatting, so as not to confuse people like me. :cool:

 

[mmm4444bot]

… start with the 'highest' exponent and compute down.

Ha. My brain woke me up at 2:30 this morning, to tell me that, when you wrote "highest", you literally meant height above the baseline in appearance. I had read it as "largest". So, please ignore my previous comment. :cool:

 

[Beowulf]

Ha. My brain woke me up at 2:30 this morning, to tell me that, when you wrote "highest", you literally meant height above the baseline in appearance. I had read it as "largest". So, please ignore my previous comment. :cool:

No problem, in context it was clear what you meant and the source of the confusion.