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Let's look at a series of base with add 1 positive exponent

5^1 = 5, 5^2 = 25 5^3 = 125, etc.

On the other side, we can also add 1 negative exponents:

5^(-1) = 1/5, 5^(-2) = 1/25, 5^(-3) = 1/125, etc.

So we have two series

5, 25, 125... and 1/5, 1/25, 1/125, ....

(The dots mean that you could continue the series as long as you wanted.)

In the first series you are multiply with 5 and in second series you are dividing with 5

If we add 1 as a sort of mathematical glue between the two series, we get

... 1/125, 1/25, 1/5, 1, 5, 25, 125, ...

Now this above series is much better than the above two series.

In order to make this series, we need 1... which is corresponds to 3^0. So can also write the above series as

... 5^-3, 5^-2, 5^-1, 5^0, 5^1, 5^2, 5^3, ...

Now why is anything raise to power 0 equal to 1?

Consider a^5/a^5 . i.e 5 a ----- = 1 5 a As you know this is the same as (a*a*a*a*a)/(a*a*a*a*a) = a^0

So to get the result we subtracted the powers to give 5-5 = 0 But we know that a^5/a^5 = 1, and so a^0 = 1

This does not depend only on a, and is true in the general case. Therefore it is true that "Anything raise to power zero is equal to 1"

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1/3 -2/3 2n - 2n -------------- = ? -2/3 2n

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