Negative Exponent (Math Question)

The pattern is easier to see as division rather than multiplication, perhaps.

Every time the exponent gets bigger, you're multiplying the number by itself once again. So,
3^4 = 3 x 3 x 3 x 3 = 81 (Sorry, can't find an alt code!)
3­³ = 3 x 3 x 3 = 27
3² = 3 x 3 = 9
3¹ = 3

Conversely, then, each time you lower the exponent, you're *dividing* by the base number.
81 ÷ 3 = 27 (which is 3³)
27 ÷ 3 = 9 (which is 3²)
9 ÷ 3 = 3 (which is 3¹)

So going from
3¹ = 3
3º = 1 (which would be 3 ÷ 3)
And when you have negative exponents:
3^-1 = 1 ÷ 3 (or 1/3)
3^-2 = 1/3 ÷ 3 (or 1/9)

Hope this helps :-)

ETA: Ha! Ninja'd several times while I was looking for alt codes, lol! To me, they're easier to read, though :-)

@ Kristen - "because someone said so a long time ago" reminds me of 4th grade, when my teacher told us, "You can't divide by 0. You can't. It's impossible. Don't even try." And I wasn't one to question stuff like that, lol! Wasn't until I was volunteering in my kid's classroom, and *seeing* how division was done with math manipulatives, that I understood why!

This all comes from the rules of exponents. For example, when you divide two numbers with the same base, you subtract the exponent in the denominator from the one in the numerator. That yields your final exponent. For instance, 2^3/ 2^3 = 2^3-3 which is 2^0. If you divide ANY number by itself, the final solution is one. Hence, anytime you're dividing two numbers with the same base and the same exponent they cancel out and the final answer will be one. Using the rules of exponents, you would subtract the exponents and obtain an answer of zero.

I hope this helps.

It is part of a bigger picture. If you had a^5, that would be a*a*a*a*a 5 times. If you had (a^5)/(a^3), that would be (a*a*a*a*a)/(a*a*a). So there would be 5-a's in the numerator (top) and 3-a's in the denominator (bottom). If each a on top can cancel one a on the bottom, then 3-a's will cancel. On the top you'd still have 2 a's left. Or, you could remember that you can subtract the powers, 5-3=2, so you have a^2.

If you flip that fraction and have (a^3)/(a^5), that would be
(a*a*a)/(a*a*a*a*a) and after canceling there would only be 2-a's on the bottom. When you cancel, and there's nothing left, there's really 1. So the simplified answer would be 1/(a^2). Since 3-5= -2, the answer can also be written as a^(-2).

I actually tell my class that a negative power means "move me and make me positive." If it's in the numerator, move it to the denominator and make the power positive. If it's in the denominator, move it to the numerator and make the power positive. If the entire fraction being raised to a negative power, flip it and make the power positive.

This is so much easier to explain on a board! (chalkboard, dry erase board, smartboard, take your pick :-)

I can email you my class notes, if you think that would help.

My head hurts. Now I remember why I used to drink so much in college. ;)

Oh, Ephie--you sure know how to live!
Good luck!
LOL

One of the most important things that helped me be successful in math as an adult (I failed math in high school and HATED it) is that you just have to accept certain things for what they are - especially if it is basic mathematical principles.
Many of these principles can be proven by mathematical proofs... but when you first learn them you do not yet have the skills to understand those proofs. For someone like me, who always wants to know how things work and why they are the way they are this is very frustrating and was the root of my difficult relationship with math.

So take it from me: math is all about following rules and recognizing patterns. Do not try to to figure out the how or why! It will drive you nuts and discourage you! For now anything to the power of 0 equals one. Period.
One you get into higher math these things will make a lot more sense - but for now just accept them the way they are.

Good luck!

Just M got it right. 7^0 is =1, not 7 (sorry Jo). 7^1 = 7.

To make Just M's answer a tad bit clearer
3^(-3), 3^(-2), 3^(-1),3^0, 3^1, ... are the same as
1/(3^3), 1/(3^2), 1/(3^1), 1/(3^0) or just 3^0, 3^1, ... which are
1/27, 1/9, 1/3, 1, 3, ...

Why? Because someone said so a long time ago, and it stuck. It's just a tool. There is no mathematical proof for this one.

This site has a great explanation though:
http://www.homeschoolmath.net/teaching/negative_zero_expo...

because the numerator and denominator would be the same causing the answer to be one

heres the answer that works best for M. from a math site
http://mathforum.org/dr.math/faq/faq.number.to.0power.html

So what is the pattern in the bottom sequence? Well, every time you move to the right in the list you multiply by 3, and every time you move to the left in the list you divide by 3. So we could take the bottom sequence and keep going to the left and dividing by 3, and we'd have the sequence that looks like this:
..., 3^-3, 3^-2, 3^-1, 3^0, 3^1, 3^2, 3^3, 3^4, ....

..., 1/27, 1/9, 1/3, 1, 3, 9, 27, 81, ....

Alright I am wicked good at math but I don't understand the question.

A number to the zero or no power is the number, not one. So seven to zero power is seven. It is not like multiplication it is power of....

Okay I have been drinking wine trying to get used to bi focal contacts so I may be wrong here. I will try to remember to come back when I can see straight. :(

I am pretty sure I am right so three to the third power is 27?

I need to rest my eyes.

Kristen, hush, it is good wine. :p Yeah, yeah I am really in no state of mind for math and I already made poor M explain it to me.

Ephie so far as my poor brain can explain it comes down to simple multiplication and division.

I feel ya, sorry I don't have any tricks, but I am also taking math this summer. My school uses Mathzone for online students, good luck!

It's great that you are looking to understand the math and not just accepting the rules! When people simply look for patterns and memorize rules, they don't have the foundation to understand newer, more complex concepts. Sorry Ina - it sounds like you had terrible math teachers! I think the other mama's already explained this one. In defense of Kristen's response, these are symbols we invented to manipulate written numbers and there are no proofs. "Someone said so" sounds bad but we are talking about a convention...