Negative or zero exponent
Why is 20 = 1? And what does a negative exponent mean?
Present several of these kind of lists to your students and ask them to find the pattern in them:
The pattern is that at each step going down, you divide by the same number, either by 2, 3 or 10. This automatically leads to the facts that 20 = 1, 30 = 1, and 100 = 1. We could do the same process for other numbers too and it would work the same way. So at least for all positive whole numbers a it is true that a0 = 1
The video below shows the same idea: teaching zero exponent starting with a pattern. This justifies the rule and makes it logical, instead of just a piece of "announced" mathematics without proof. The video also shows another idea for justifying this: we can multiply powers of the same base, and conclude from that what a number to zeroth power must be.
Zero Exponent video
You can try if the same works for negative numbers (negative base) - and it does! You just divide by that negative number at each step down (here by negative 2):
(-2)4 = 16
(-2)3 = -8
(-2)2 = 4
(-2)1 = -2
(-2)0 = 1
Next continue the same pattern even more - and you'll enter the negative exponents. Write down the following table without the answers, and ask your child to complete it. When done, ask if she notices any patterns.
divide by 10
103 = 1000
After you have noted the patterns together, ask your child to continue the tables somewhat in both directions. Also ask her to do on her own one with number 4 and with -2.
Then, let's look at the columns a little more closely.
Based on this observation, ask then your child, what about 2-5 ? The answer we want is of course that 2-5 = 1/25. Similarly, 4-7 = 1/47. This observation is usually stated as the definition for negative exponents, and now you know where it came from! In other words, the typical definition says to take the reciprocal of the base number, and raise it to the corresponding positive power.
The video below also explains this same idea: teaching negative exponents based on a pattern.
Negative Exponents: Learn Them with a Pattern!
|What about if the base is negative? How is (-3)-2 done?
Follow the same principle as above. Divide by -3 at each step.
Or, you can use the definition:
(-3)-2 = 1/(-3)2 = 1/9
(-3)-3 = 1/(-3)3 = 1/(-27) = -1/27 etc.
|Are there any games you can play with the concept of negative exponents once the kids have grasped it?
I have not found any games on this on the internet (let me know if you have one!). Here are some simple suggestions for a game to make at home
Finding a negative power of a fraction is not something you'd necessarily present first time studying negative exponents. But here go:
You can again look for a pattern when charting the different powers of 1/5.
You will see that each successive step down is like multiplying by 5. Why? Because if you follow the idea above, you would be
So dividing by 1/5 is the same as multiplying by 5 Continuing the chart:
Or, you could use the definition, which says to take the reciprocal of the number raised to the positive exponent. So for example,
Why is the answer to any digit with the exponent of zero equal one. (The theory not just because it is)
Please see Two proofs that a number to zero exponent is one. Actually you can also plain DEFINE n0 to be 1, and then give the "proofs" presented in that article as justifications why we should define it so - it is the only reasonable definition that makes it work with the laws of exponents.
Please see also n^0 Power = 1: Defined or Proved?
Are Negative Exponents Like Other Exponents?
Is there a general rule for doing all exponents, or does a negative exponent have nothing in common with positive exponents?
n to 0 power
Why any number raised to the zero power is equal to one.
Can you have exponents that are decimals?
Meaning of Irrational Exponents
But where do irrational exponents fit in? Can you raise 2 to the sqrt(2) power? Is there any definition for this?