How to simplify and write answer using negative exponent ?

( -5 )  ^  -2  /  -5 ^ 3

Are both bases negative 5 or is base on the numerator negative 5 and base on denominator
5 ?
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Adding parenthesis to clarify we have

 ( ( -5 ) ^  (-2)  ) /  ( (-5)  ^ 3 )

So giving

1/(-5)^5 = -1/5^5

This is the convention used to break the ambiguity in these expressions. The minus on its own is always applied before all other mathematical operations.
I think the bottom line is that it is ambiguous. Even if there is a formal definition, it is incorrectly interpreted sufficiently often that you should just avoid expressing negative exponents that way without clarifying with parenthesis.

See here:

and here:

These papers argue exactly against what GwynforWeb just said, and say that order of operations is:

1) Parenthesis
2) Exponents
3) Multiplication and/or division
4) Addition and/or substraction

And that the negative sign falls under the addition/subtraction part of orders of operation. So if you take your denominator for example:

-5 ^ 3

They suggest that the minus at the beginning is ACTUALLY shorthand for saying this:

0 - 5^3

Which means it is interpreted like this:


However, if I go into Microsoft Excel and type two formulas:

     = -(5^4)           -->    -625
     = -5^4             -->     625
     = (-5)^4           -->     625

These findings contradict the papers I just posted above and agree with GwynforWeb.

The fact that there are such discrepancies tells me the only real answer is you simply have to simply avoid that notation and use parenthesis to keep things unambiguous.
naseeamAuthor Commented:
Thanks for adding parentheses for clarification.

The bases are different so how can you add exponents ?
1/ ( (-5) ^ 3 ) = (-5)^(-3)         (  Using rule: 1/(x^m)= x^(-m)  )


 ( ( -5 ) ^ (-2)  ) /  ( (-5)  ^ 3 )

= ( ( -5 ) ^ (-2) ) ( (-5) ^ (-3) )

Both terms are -5 raised to a power, now just add the powers, giving

 ( ( -5 ) ^ (-2) ) ( (-5) ^ (-3) ) = ( -5 ) ^ (-5)              (  Using rule: x^m*x^n = x^(n+m)  )

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Frosty555, you are correct this is a complete mess. Order of precedence is confusing for most at the best of times. The minus is being used as a unary operation and as such has higher preference over binary +,-,* and /. The teacher should have been clearer in asking the question. Conventions in computer expressions are well defined but in written maths we make short cuts. Writing 1/-5³ is sloppy and ambiguous in every sense and the teacher should have been more careful. My guess is, and I am guessing, he or she meant 1/(-5)³
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