Math CentralQuandaries & Queries


Question from Michael:

I was teaching my student about exponent when we stumbled across this problem.

"write (-125)^-3 in exponent form"

The problem is with the number "-125". I think -125 = (-5)^3. But one of my student thought that it is -125= -5^3.
Both of them is equal to -125 but they totally different in structure. I wonder which one is correct and why it is. I am worried if they get this wrong, it might spell problem in the future when we start substituting the numbers with variable.
Thank you very much for reading my question and answering it.

Hi Michael,

Both of your expressions are correct,

\[(-5)^3 = -125 \mbox{ and } -5^3 = -125.\]

The problem here is the term "exponential form". Exponential form means any form involving exponents. Hence both your two expressions are correct, but then $(-125)^3$ is also in exponential form since it uses an exponent. I know that what is expected by the question is to have you use the fact that $5^3 = 125$ and then use the rules of exponents to rewrite the resulting expression. The problem is the wording of the question "write $(-125)^{-3}$ in exponential form". Maybe the question should be worded "simplify $(-125)^{-3}",$ but even here I am not sure what the "correct" answer is. Not only is there your conundrum with your student but there is also a negative exponent. Do you leave the final expression with a negative exponent or do you express it as a fraction, with an expression in the denominator which has a positive exponent.

It is not true that every problem in mathematics has one unique correct answer. The object of this problem, whichever way it is worded, is to have your students use the rules of exponents correctly and arrive at a valid expression.

I hope this helps,


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