Proof - Any real positive value raised to the power of zero is 1?

Question

G'day,

I was wondering what sort of proofs everyone has seen regarding this problem?

Cheers,

David

note - Ive proven it with limits but was wondering if there was an alternative strategy

Answer 1

There is.  Using the properties of exponents, you know that

a^m/a^n = a^(m - n)

You also know that any number (other than zero) divided by itself is equal to 1.

So, a^7/a^7 = 1

and,

a^7/a^7 = a^(7 - 7) = a^0 = 1

(for all 'a' not equal to zero.)

Do this a few times...

QED

(By the way, this is true for any non-zero base, not just real positives.)

Answer 2

x^y = exp(y*ln(x))

So x^0 = exp(0*ln(x))

=exp(0) = 1

Answer 3

a^m / a^n = a^(m-n)

in ase of m=n

a^m / a^m = a^0

a^m ... a * a * a * a * ... * a  { m times}

-----  = ----------------------------

a^n  ... a * a * a * a * ... * a  { m times}

=1 by simply the term