# Laws of Exponents

The “laws of exponents” are a frequent source of errors.  The rules that apply when simplifying expressions that involve exponents can be figured out quickly on your own if you (in your mind’s eye) expand integral exponents into repeated multiplication. I encourage students to master being able to explain why each of these rules is as it is instead of memorizing them, as memorized versions are more likely to get jumbled together in your thinking when working problems.

The answer shown below is wrong.  Try working your way through the problem backwards, from the answer up. As you find each mistake, try to identify the thinking behind it… what perspective led to the mistake? What should have been done instead?

$\dfrac{(ab^2)^3(5a^2)}{a^2b}$

$=\dfrac{a^5b^525a^2}{a^2b}$

$=\dfrac{25a^{10}b^5}{a^2b}$

$=25a^5b^5$.

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Hint 1: The correct answer to the original problem is: $5a^3 b^5$

Hint 2: Several mistakes have been made in the work above – don’t just seek the correct answer – look for the mistakes between every pair of lines.