Teaching negative exponents

Continuing on with the exponents, I have another video about negative exponents. Check also the "prelude" to this, my video on zero exponent.


Negative Exponents: Learn Them with a Pattern!


In books we usually find the definition that x−n = 1/xn or perhaps x−n = (1/x)n (you take the reciprocal of x and use the opposite of the exponent).

But again, I want to highlight that we DON'T have to just "announce" to students this definition or how negative exponents are done. We can justify it. In the video, you can see this done first with the help of a pattern, where we go from the positive exponents to zero exponent to negative ones.

I also show another way of showing your students why negative exponents have been defined the way they are. If you look at the shortcut for dividing powers with the same base (I used 63/65 in the video), we can simplify as usual and arrive at 1/62. However, the shortcut says we can just subtract the exponents, so 63/65 also equals 6−2. The same reasoning applies for any non-zero x and any positive whole number exponent.

This way, we teach our students not only the "how" but also the "why" of mathematics, and help them become better thinkers. Mathematics is not just about memorizing things and then learning to "spit" them out at tests - it also has a LOT to do with proof and logical thinking. And surely our kids need that today, more than ever.
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