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Negative or zero exponent
A teaching guideline/lesson plan with some ideas on how to teach this topic

Why is 2⁰ = 1? And what does a negative exponent mean?

Present to your child several of these kind of lists and ask her to find the pattern in them:

2⁵ = 322⁴= 162³= 2²=2¹=2⁰=

3⁵ = 2433⁴= 813³= 3²=3¹=3⁰=

10⁵ = 100,00010⁴= 10,00010³= 10²=10¹=10⁰=

The pattern is that at each step going down the list 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

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)⁵ = ^-32
(^-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.

At every step
divide by 2

2³ = 82²= 42¹= 2 2⁰= 1
2^-1= 1/22^-2= 1/4
2^-3= 1/8

At every step
divide by 3

3³= 27 3²= 93¹= 33⁰= 1
3^-1= 1/33^-2= 1/9
3^-3= 1/27

At every step
divide by 10

10³= 1000 10²= 10010¹= 1010⁰= 1
10^-1= 1/1010^-2= 1/100
10^-3= 1/1000

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.

At every step
divide by 2

2³ = 8
2²= 4
2¹= 2
2⁰= 1
2^-1= 1/2
2^-2= 1/4 = 1/2²
2^-3= 1/8 = 1/2³

At every step
divide by 3

3³= 27
3²= 9
3¹= 3
3⁰= 1
3^-1= 1/3
3^-2= 1/9 = 1/3²
3^-3= 1/27 = 1/3³

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.

What about if the base is negative? How is (-3)^-2 done?

Follow the same principle as above. Divide by -3 at each step.
The sign of the answers alternates.

(-3)³ = -27
(-3)² = 9
(-3)¹ = -3
(-3)⁰ = 1
(-3)^-1= -1/3
(-3)^-2 = 1/9
(-3)^-3 = -1/27
(-3)^-4 = 1/81

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

Make problem cards and answer cards, and play a memory game.
Have one stack of cards that would give you the base, and another stack of cards that would give you the exponent. Then take any board game where you can roll the dice and advance your peg from a beginning to some kind of goal. Play so that at each turn you first take one card from the 'base' stack and one card from the 'exponent' stack, and if you answer right, then you can roll the dice and get somewhere. Or make up your own rules based on the original board game.
Or play the other way round: have a stack that has 'answer' cards, numbers like 1/9, 1/8, 1/25, 1, 4, 1/125, 1/32, 32, 16, 36, etc. and use those in the board game (some would have positive exponents).

What about if the base is a fraction? How is

(

)

^-2 done?

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.

(

)⁴ =

625

(

)³ =

125

(

)² =

(

)¹ =

(

)⁰ = 1

You will see that each successive step down is like multiplying by 5. Why? Because if you follow the idea above, you would be

dividing by

at each step. Now, you should be aware from

fraction division that if you divide any number by

, it is the same

as multiplying the number by the reciprocal of

, which is 5.

So dividing by 1/5 is the same as multiplying by 5 Continuing the chart:

(

)² =

(

)¹ =

(

)⁰ = 1

(

)^-1 = 5

(

)^-2 = 25

(

)^-3 = 125

(

)^-4 = 625

Or, you could use the definition, which says to take the reciprocal of the number raised to the positive exponent. So for example,

(

)^-2 = 5² = 25

(

)^-6 = 5⁶ = 15625

(

)^-3 =

(

)³ =

= 3

)^-4 =

(

)^-4=

(

)⁴=

2401

6561

Why is the answer to any digit with the exponent of zero equal one. (The theory not just because it is)
Jose Gonzalez

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?

More information:

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.

Decimal Exponents
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?

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Negative or zero exponent A teaching guideline/lesson plan with some ideas on how to teach this topic

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Negative or zero exponent
A teaching guideline/lesson plan with some ideas on how to teach this topic

Subscribe for free
HOMESCHOOL MATH NEWSLETTER