Exponents

[mickeybell]

okay so my math teacher just taught us exponents today but with negative numbers. i must be really dumb not understanding or it is just plan hard. anyone can help.

 

[mickeybell]

also it had letters like;
x^{-8
3x-5
also like,
3x0
------
y-3}:twisted::twisted::twisted::twisted::twisted::twisted::-?:-?:idea::idea:

 

[JeffM]

okay so my math teacher just taught us exponents today but with negative numbers. i must be really dumb not understanding or it is just plan hard. anyone can help.

You are not dumb, nor is it hard. It's just new.

\(\displaystyle a\ is\ a\ real\ number \ne 0 \implies a^0 \equiv 1.\) That is a definition.

\(\displaystyle a\ is\ a\ real\ number \ne 0\ and\ n\ is\ an\ integer > 0 \implies a^n \equiv a * a^{(n - 1)}.\) That too is a definition.

\(\displaystyle a\ is\ a\ real\ number \ne 0\ and\ n\ is\ an\ integer > 0 \implies a^{-n} \equiv \dfrac{1}{a^n}.\) That too is a definition.

Those three definitions completely DEFINE what we mean by an exponent that is an integer for a number that is not zero.

So \(\displaystyle 3^1 = 3 * 3^0 = 3 * 1 = 3.\)

\(\displaystyle 3^2 = 3 * 3^1 = 3 * 3 = 9.\)

\(\displaystyle 3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}.\)

That's it. It's one of those things that is so simple that you say, "Oh, there must be something I am missing here because how can they make a fuss about that? It must be super hard and I just can't get it." Nope. It really is that simple.

Now give those problems a shot, post the answers, and let us see how you did.

PS Be careful with \(\displaystyle (3x)^{-5}\ versus\ 3x^{(-5)}.\) They have different answers. Do you see why?