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We have 64^ -1 = 1/64 But 1/64 can be written as an integer raised to an integer power in other ways, too! How many different ways in total can1/64 be written as an integer raised to an integer power, including 64^ -1?

 May 9, 2020
 #1
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We have 64^ -1 = 1/64 But 1/64 can be written as an integer raised to an integer power in other ways, too! How many different ways in total can1/64 be written as an integer raised to an integer power, including 64^ -1? 

 

64–1     8–2     4–3     2–6    

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 May 9, 2020
 #2
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I already tried that. it isn't right.

ss333  May 9, 2020
 #3
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You tried 4 and were told it isn't right.  Tried it where?  Who says it's not right?  I'm not accepting that, but, just for argument, let's assume your source is correct.  We know the above four work, so the answer can't be fewer, it has to be be more.  Try 5, and then if necessary 6, 7, 8, etc. until you come to the number that is considered right.  Give us that clue.  I cannot think of any different ways to make the required value, limited to using an integer base and an integer exponent.     

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Guest May 10, 2020
 #5
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Who gave me that negative heart?  I haven't seen YOU contributing to the solution.  I understand now how I didn't get it right the first time - an oversight, obvious in hindsight - but I stuck with it and kept coming back and I think I finally got the correct answer.   

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Guest May 10, 2020
 #4
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"I already tried that. it isn't right."

 

It's me again.  I had an idea ... and it's so crazy it just might work. 

 

It occurred to me that when you have a negative number to start with,

raising it to an even-numbered exponent produces a positive result. 

 

So here's my idea:  Try 6 and see if that gets accepted. 

 

If it works, it's because these are the other two:  (–8)–2  and  (–2)–6  

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 May 10, 2020

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