Pre-Algebra question about exponents

[Daisy]

I'm not even sure I can really explain this without using superscripts but here goes.

 

According to my daughter's pre-algebra book...

 

Any number (that does not equal zero), multiplied by a zero exponent, equals one.

 

Huh?

 

Why doesn't it equal zero?

 

It seems to be that

 

2 squared is four.

 

2 to the one power is two.

 

Wouldn't 2 to the zero power be zero? They say it equals one and neither of us is understanding the reasoning behind that.

[Jen500]

I always remember this by looking at the pattern-

 

2 to the fourth power=16 (16/2 = 8)

 

2 to the third = 8 (8/2 = 4)

 

2 squared = 4 (4/2 = 2)

 

2 the the first= 2 (2/2 = 1)

 

2 to the 0 power= 1

[Carol in Cal.]

Any number (that does not equal zero), multiplied by a zero exponent, equals one.

 

Wouldn't 2 to the zero power be zero? They say it equals one and neither of us is understanding the reasoning behind that.

 

First of all, it's not multiplied by a zero exponent. It is RAISED to a zero power. Multiplication by zero always equals zero, which is why I'm nitpicking you on this point. If you stop talking about multiplication, it is easier to accept that the zero power does not have to equal zero.

 

Secondly, it goes back to the multiplicative identity. Anything multiplied by one equals itself. Anything multiplied by zero equals zero. Saying that something is multiplied by one does not change it. That is the effect you want from something being raised to the zero power. That's how I always remember it.

[Deniseibase]

This might help -

 

http://www.khanacademy.org/video/zero--negative--and-fractional-exponents?playlist=ck12.org%20Algebra%201%20Examples

[8filltheheart]

I'm not even sure I can really explain this without using superscripts but here goes.

 

According to my daughter's pre-algebra book...

 

Any number (that does not equal zero), multiplied by a zero exponent, equals one.

 

Huh?

 

Why doesn't it equal zero?

 

It seems to be that

 

2 squared is four.

 

2 to the one power is two.

 

Wouldn't 2 to the zero power be zero? They say it equals one and neither of us is understanding the reasoning behind that.

 

Agree w/Carol's explanation as to why it doesn't equal zero. But why does it equal 1? I didn't watch the video, so maybe it explained it simply, but here is my explanation in case it is different.

 

Any number raised to the 1st power equals itself. (so 2^1 = 2) The fraction 2/2 is the same as 2^1 divided by 2^1 (do you see that b/c the 1st power means that is same as having 1 of that number) Well, it also is true that when you move a number raised to a power to the numerator or to the denominator that it becomes its opposite......so 1/2^1 is the same as 2^-1.

 

You know that 2/2=1 Correct?

 

Numbers with the same bases being multiplied, you add the exponents. FOr example 2^2 * 2^3= 2*2*2*2*2 or 2^2+3 or 2^5

 

So, 2^1 * 2^-1= 2^1-1 or 2^0=1

 

HTH

[JudyJudyJudy]

Khan Academy's first explanation is good, though as a teacher, I always did a lot of examples to get the point across.

[Dana]

Explanation also halfway down the page here on purplemath.

 

I show it using exponential rules:

If you are okay with a^m * a^n = a^(m+n)

then to avoid contradiction, a^0 = 1 (as long as a isn't 0)

 

2^0 * 2^1 = 2^(0+1) = 2^1

 

so

(?)(2) = 2

 

The only thing that'll work is if ? is 1,

so 2^0 has to be 1.

[Crimson Wife]

Maria Miller also has a great video explaining it here.

[Daisy]

Thanks, ladies. I"m going to watch these videos and look at the sites. Daughter had a major meltdown over this yesterday and I'm going to try to figure it out better before tackling it with her again.

 

Can you think of any reason why the book introduced this concept on its very FIRST lesson on exponents? Part of me just wants her to move on and wait until she is more familiar with exponents and then revisit it. Does anyone see a major problem with that?

Edited August 24, 2011 by Daisy

[JudyJudyJudy]

Daisy, some people may disagree with me on this, but I wouldn't stress over her understanding it right now. Sometimes understanding comes later. It's okay for her to memorize the rule now, and then hopefully the understanding will come later.

[JudyJudyJudy]

I forgot to address the second part: I think the book is doing the right thing by introducing this early. We need to know the rules of exponents in order to work with exponents.

[Sebastian (a lady)]

I'm not even sure I can really explain this without using superscripts but here goes.

 

According to my daughter's pre-algebra book...

 

Any number (that does not equal zero), multiplied by a zero exponent, equals one.

 

Huh?

 

Why doesn't it equal zero?

 

It seems to be that

 

2 squared is four.

 

2 to the one power is two.

 

Wouldn't 2 to the zero power be zero? They say it equals one and neither of us is understanding the reasoning behind that.

 

I agree with making the distinction that you are raising the number to a power, not multiplying it by the exponent.

 

The reason A^0 = 1 is that an exponent represents the number of times that the base number (A) is multiplied by itself. So A^2 = A*A and A^3 = A*A*A.

When you multiply A^2 * A^3 you are doing this: A*A * A*A*A . So you get A^5. You can write this rule as A^b * A^c = A^b+c.

 

When you divide exponents

A^6 / A^2 = A*A*A*A*A*A / A*A

You can cancel numbers that appear in both the numerator and the denominator. So you end up with A*A*A*A.

The rule that follows is A^b / A^c = A^b-c.

 

When you write A^0, what you are saying is that the number of times the base number (A) was in the numerator and denominator were equal. (Ex. 2^2/2^2 or 4/4.) This is the same as saying that A^b/A^b = A^b-b = A^0.

 

Because we know that 4/4 or A/A or A^93/A^93 all are the same as one whole, we can say that A^0 (which represents a situation where the top and bottom of the fraction are the same) are equal to one whole, or just 1.

 

I hope this helps.

 

One one hand, it is something that you could just learn. On the other hand, it is a natural outcome if you understand what the rules of exponents are showing. So playing with it a little will help to solidify other understanding of exponents.

 

FWIW, 0^0 is undefined. On one hand, you have the same number on the top and bottom of the fraction, suggesting that it would be 1. On the other hand, you have the rule that you cannot divide by zero.

 

AOPS has a great chapter on exponents, though it will make your head throb to do the gymnastics involved in the exercises (a good hurt, like doing a good workout).

[kiana]

Because if it were anything other than 1, the law of exponents that says a^m times a^n = a^(m+n) would be inconsistent.

 

You can come up with many examples where you can see that it must be so. Anything where m+n = 0 (so negative exponents), or anything where n = 0 ...