why is any number to the power of zero; one?

Question

hello! Why is anything to the power of zero = one? What rule is this?

"Simple "

ok what rule is it? please do tell.

....and the best answer goes to rfamilymember....thank you to all who have tried to answer and tried to participate. bye now!

I hope you at least enjoyed your two points for answerings so you don't feel as bad.

Answer 1

x^0 can be written as x^(n-n)=x^n/x^n by law of exponents=1

Answer 2

A basic rule of exponents is

---x^n * x^m = x^(n+m)

Using n=0 we'd have...

--- x^0 * x^m = x^(0+m) = x^m

In this case, the number x^m times x^0 gets you the same number x^m. The only multiplier that preserves the number in this way (assuming x isn't 0) is 1. Hence, x^0 is 1, which would be the case for all x not 0.


Of course, that leaves 0^0. One could make an argument for it being either 0 or 1. By convention, we generally agree it to be 1, as it makes a bunch of other formulas and such much, much easier to follow and use when 0 is involved.

Answer 3

Simple.

You should be familiar with the rule for dividing powers, ie:

x ^ 5 / x ^ 2 = x ^ (5 -2) = x ^ 3 (x is any number)

So consider:

x ^ 6 / x ^ 6 (you could use any number here)

by the same rule as above:

= x ^ (6 - 6)

= x ^ 0

but x ^ 6 / x ^ 6 = 1

so x ^ 0 = 1

This works with any numbers - try it

Answer 4

psycho_jelly_babies has it right, although I do want to add that you can't do it with *any* number -- specifically, taking zero to the zero power (0^0) is not allowed. To see why, follow his logic, but use 0 for the base... you'll quickly find that you have zero in the denominator of a fraction, which, of course, is prohibited.

Answer 5

Count backward with your exponents from 3. Ready?
2³ = 8. Half of that is
2² = 4. Half of that is
2¹ = 2. Half of that is
two to the zero power. It's 1.

You can do that for any non-zero number.
5³ = 125. One-fifth of that is
5² = 25. One-fifth of that is
5¹ = 5. One-fifth of that is
five to the zero power. It's 1.

Answer 6

It follows from the fact that

x ^ (a + b) = x ^ a * x ^ b
and
x ^ (-c) = 1 / x ^ c

Note

x ^ (0) = x ^ (1 - 1) = x ^ 1 * x ^ (-1) = x / x = 1 ; if x is not 0

Answer 7

any thing to the power is zero is zero and any number to the power of 1 is that number. for example 9^1=9

Answer 8

To understand x^0 think about

V=1*x*x*x

as the way you'd get x^3 (repeat the x as a factor 3 times)

But x^0 says don't use *any* x's as factors. That just leaves

V=1

And, BTW, 0^0 is undefined.


Doug

Answer 9

because all exponential functions (trancendent function) are continuous funtions. If you look for the values between 10^-1 and 10^1 you will see why. 10^-1 = 0.1 and 10^1 =10 You can calculate every point in between... 10^-0.5 and 10^0.5 (= root square of 10) and so on...

Answer 10

2^2 would be 2x2
2^1 would be 2x1
2^0 would be.....