Pre-Algebra Question

H

Hope Spirit

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Oct 8, 2020
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Okay, so this question is from a pre-algebra class, but I have a feeling it's a bit advanced.
This is the exact text:
We have
$64^{-1} = \frac{1}{64}.$
But
$\frac{1}{64}$
can be written as an integer raised to an integer power in other ways, too! How many different ways in total can
$\frac{1}{64}$
be written as an integer raised to an integer power, including
$64^{-1}$
?

I know how to do each step separately, but I don't know how to put it all together. Can anyone help?
Thanks,
Hope
P.S. - Yes, my name actually is Hope Spirit. My parents had a lot of fun with that.
 
Several ways...
[math]64 = 8^{2}[/math][math]64 = 4^{3}[/math][math]64 = 2^{6}[/math]
 
Hope Spirit said:
Okay, so this question is from a pre-algebra class, but I have a feeling it's a bit advanced.
This is the exact text:
We have
$64^{-1} = \frac{1}{64}.$
But
$\frac{1}{64}$
can be written as an integer raised to an integer power in other ways, too! How many different ways in total can
$\frac{1}{64}$
be written as an integer raised to an integer power, including
$64^{-1}$
?

I know how to do each step separately, but I don't know how to put it all together. Can anyone help?
Thanks,
Hope
P.S. - Yes, my name actually is Hope Spirit. My parents had a lot of fun with that.
Click to expand...
Don't marry anyone from the Less family.
 
Can you tell me what that means? Also, thank you for the answer. I can't believe it was right in front of my face the whole time! I knew how to put the pieces together! I guess this is why we have math forums. Thanks again!
-Hope
 
Oh! I get it now. But I'm not really old even to be thinking about that...Not that I don't...;)
That would be hilarious though.
 
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