Question

How do you find the base when you only know the exponent? Is there a rule I don't know?

Say you have something like this:

r^8 = 6561

How would you find r?

Answer 1

You would take the 8th root of both sides...to isolate r

r^8 = 6561

\[\sqrt[8]{r}^8\] = \[\sqrt[8]{6561}\]see how that eliminates the ^8 power?

Answer 2

take log on both sides and bring down the exponent
8 log(r) = log (6561)
log (r) = log (6561) / 8
r = 10^(log(6561) / 8)

Answer 3

ah, of course @johnweldon1993 approach is easier

Answer 4

Yes but BOTH ways are a correct way of doing this :)

Answer 5

Didn't understand much but maybe I'll get it later. Thanks!

Answer 6

well, what didn't you understand?

Answer 7

I didn't really understand how you got to \[8\sqrt{6561}\] and I'm also not sure how to find the actual number from that.

Answer 8

\[r^2 = 9\]
now take the square root on both sides
\[\sqrt{r^2} = \sqrt{9}\]
square root and square cancels, so you have
\[r = \sqrt{9}\]

Answer 9

Hmm but why \[r^{2}\] and not
\[r ^{8}\]

Answer 10

it was just an example, and because square root is easier when giving an example (like sqrt(9) = 3). But it is the same principal.

Answer 11

Yes, but \[\sqrt{r ^{8}}\] wouldn't be r, right? :/ that's the bit that confuses me.

Answer 12

Correct,
\[\sqrt{r^8} \neq r, but \sqrt [8]{r^8} = r\]

In my example I used r^2 not r^8 just because it is easier.
But if you have something your square e.g.
(r^2) = 100, then to find r you will need to do the inversion which is taking the root, in this case (just an example)
\[r = \sqrt{100} = 10\]

I hope I did not confuse you

Answer 13

So does that apply to all radical problems of that form? I mean ->
\[\sqrt{a ^{b}} = \sqrt[b]{a ^{b}} = a\] all the time? I forgot lots of basic rules...

Answer 14

no that is not correct, squarertoot is not the same as the b'th root.
\[3^2 = 9 => \sqrt{9} = 3\]
\[3^3 = 27 => \sqrt[3]{27} = 3\]
but
\[\sqrt{27} = 5.19615...\]

Answer 15

This is one of those things that will probably take me ages to get ><

Answer 16

I'm sure you will get it with a bit of practice.
Maybe just remember
10 SQUAREed = 100, the SQUAREroot of 100 is 10.
Nah, that probably doesn't make it any easier