Question

Why 0 to the power of any number=1

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Answer 1

Answer 3:

Any number to the zero power always gives one.

One rule for exponents is that exponents add when you have the same base. So if you have a number, x, and exponents, a and b, then:
x^a * x^b = x^(a+b)
So then if we make one of the exponents negative:x^a * x^-b = x^(a-b)
And if the exponents are the same magnitude (a = b)x^a * x^-b = x^a * x^-a = x^(a-a) = x⁰

Now, remember that if you have a negative exponent, it means you have one divided by the number to the exponent:

x^-a = 1/x^a
So, we can also write x^a * x^-a in a different way:
x^a * x^-a = x^a * 1/x^a = x^a/x^a
And a number divided by itself is always 1 so:
x^a * x^-a = x^a* 1/x^a = x^a/x^a = 1:
So now we've shown that:
x^a * x^-a = x^(a-a) = x⁰
and
x^a * x^-a = x^a * 1/x^a:
This means that any number x⁰ = 1.

If you had trouble understanding it all with variables, let's look at it again,but this time as an example with numbers:

If we plug in numbers, (for example let x = 5, a = 2, and b = 4) then:
One rule for exponents is that exponents add when you have the same base.
5² * 5⁴ = 5⁽²⁺⁴⁾ = 5⁶ = 15625
So then, if we make one of the exponents negative:
5² * 5^-4 = 5^(2-4) = 5^-2 = 0:04
And if the exponents are the same magnitude:
5² * 5^-2 = 5^(2-2) = 5⁰

Now, remember that if you have a negative exponent, it means you have one divided by the number to the exponent:
5^-2 = 1/5² = 0:04
So we can also write 5² * 5^-2 in a different way:
5² * 5^-2 = 5² * 1/5² = 5²/5² = 25/25

And a number divided by itself is always 1 so:
5² * 5^-2 = 5² * 1/5² = 5²/5² = 25/25 = 1

So now we've shown that:
5²*5^-2 = 5^(2-2) = 5⁰
and
5² * 5^-2 = 5²/5² = 1
This means that 5⁰ = 1.

This works for any number x that you want to plug in except for x = 0,because 0/0 is indeterminate (it is like dividing zero by zero).