Arithmetic with Polynomials and Rational Expressions exponents Exponents

Negative Exponents with Tons of Variables

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I think the temptation I have is to call this a “careless” mistake and urge more practice. Let’s probe deeper.

1. What does this kid know and understand about exponents?

2. What’s the fastest way to help?

3. What makes this mistake so tempting?

Thanks to Sadie Estrella for the awesome addition to our ever-mounting pile of exponents mistakes.

3 replies on “Negative Exponents with Tons of Variables”

This student realizes that a negative exponent simplifies to the reciprocal. However, the student does not realize that the exponent is distributed to all parts of the base (when the base contains multiplication). Finally, the student knows that when simplifying exponents the exponent in the denominator is subtracted from the exponent in the numerator. But they do not know that the rule only applies when the exponents are the same.

There’s the mistake of not cubing the coefficients, there’s the mistake of “dividing out” x’s on the top with y’s on the bottom, and there’s the mistake of simplifying n^6 / n^3 to n^3 / n. That’s a lot of mistakes for a short exercise!

The kid clearly understands that negative exponents mean to take the reciprocal, so that’s good.

For the rest, I’d encourage writing out the cubing and simplifying longhand: that is, write three copies of the fraction and multiply them, write out y*y*y*y*y*y instead of y^6, and so on. That’s generally my first fallback whenever I see kids struggling with following the exponent rules correctly.

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