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# Negative or zero exponent

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

Students can discover the answers to these questions on their own! Simply present them the lists below, and ask them to find a 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 you divide by the same number at each step, 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 (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 tables without the answers, and ask the students to complete it.  When done, ask if they notice 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 = 100
101 = 10
100 = 1
10−1 = 1/10
10−2 = 1/100
10−3 = 1/1000

After you have noted the patterns together, ask the students to continue the tables in both directions.  Also ask them to write 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 the students, 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 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.
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 games you can make yourself. Make problem cards and answer cards, and play a memory game. Have one stack of cards that gives you the base, and another that gives you the exponent. Use any board game where you roll the dice and advance your piece to some kind of goal. At your turn, take one card from the 'base' stack and another from the 'exponent' stack, and simplify (solve) the resulting expression. If you answer correctly, you may roll the dice and move. Or, you can make up your own rules based on the original board game. Play the other way around: have a stack that has 'answer' cards, numbers such as 1/9, 1/8, 1/25, 1, 4, 1/125, 1/32, 32, 16, 36, etc. and use those in the board game. If the player makes up a correct expression in the form ab, the player can roll the dice and move. Some or all of these answers can be based on positive exponents. This is a two-person game involving negative exponents. Give each person the cards from 1 through 5 (you can use a regular deck of cards, using the ace as 1). The black cards signify positive numbers and the red cards negative numbers. One player chooses one of his cards to be the BASE and places it on the table. At the same time, the other player chooses one of his cards to be the exponent, and places it on the table. The player that calls out the correct answer first wins poth cards, and puts them into his personal stack. If it is a tie, the cards stay on the table and the players play another round. The winner of that round gets all four cards. The winner is the student who collects more cards.

 What about if the base is a fraction? How is ( 1 5 ) −2 done?

You can again chart the different powers of 1/5 and look for a pattern.

 ( 1 5 )4 = 1 5 × 1 5 × 1 5 × 1 5 = 1 625
 ( 1 5 )3 = 1 5 × 1 5 × 1 5 = 1 125
 ( 1 5 )2 = 1 5 × 1 5 = 1 25
 ( 1 5 )1 = 1 5
 ( 1 5 )0 = 1

You will see that at each successive step, you multiply by 5.  Why?

If you just took the idea from above (where we divided by the same number at each step), you'd divide by 1/5 at each step.  But, you should know from fraction division that dividing any number by 1/5 is the same as multiplying the number by the reciprocal of 1/5, which is 5.

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

 ( 1 5 )2 = 1 5 × 1 5 = 1 25
 ( 1 5 )1 = 1 5
 ( 1 5 )0 = 1
 ( 1 5 )−1 = 5
 ( 1 5 )−2 = 25
 ( 1 5 )−3 = 125
 ( 1 5 )−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,

 ( 1 5 )−2 = 52 = 25
 ( 1 5 )−6 = 56 = 15625
 ( 2 3 )−3 = ( 3 2 )3  = 3 2 × 3 2 × 3 2 = 27 8 =  3 3 8
 (1 2 7 )−4  = ( 9 7 )−4  = ( 7 9 )4  = 7 9 × 7 9 × 7 9 × 7 9 = 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 simply 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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