Question

SUPER QUICK
4z^-4 is it 1/4z^4 or 4/z^4?

Answer 1

\[4z ^{-4}\]

Answer 2

Does the 4 have a negative exponent?

Answer 3

indeed

Answer 4

So you really meant that the problem is:
\[(4z)^{-4}\]

Answer 5

not in parenthesis

Answer 6

So the 4 DOESN'T have a negative exponent.

Answer 7

\[4z ^{-4}\]

Answer 8

looks like that

Answer 9

In that case on what basis could you justify moving it to the denominator?

Answer 10

\[4z ^{-4}=4^1z ^{-4}=\frac{4}{z^4}\]

Answer 11

Thanks

Answer 12

yw

Answer 13

Rather than just blithely giving away the answer to be forgotten later, how do we know that a negative exponent is really just divided like that?

Well suppose you have 16/4, that's the same as:\[\frac{ 16 }{ 4} = \frac{ 2^4 }{ 2^2 }\]
Now of course we know that 16/4 =4 right? But now we see that we have a really nice rule where:\[\frac{ 16 }{ 4} = \frac{ 2^4 }{ 2^2 }=2^4*2^{-2}=2^{4-2}=2^2=4\]

since we already know that exponents add, just play around with exponents and you will see that 2^3*2^2=2^5. we can see that the rule is just subtraction for division. fun stuff