Why 5^0 = 1?

[brucejin]

I can explain
5^2 = 5x5 = 25
5^3 = 5x5x5 = 125

But how to explain 5^0 = 1?

Thanks

 

[mmm4444bot]

… how to explain 5^0 = 1 …

Hi Bruce:

The short explanation is, "Because we define it to be so."

Here are two more explanations.

Consider the following sequence.

3125, 625, 125, 25, 5, 1

Each number (after 3125) is obtained by dividing the preceeding number by 5.

3125/5 = 625
625/5 = 125
125/5 = 25
25/5 = 5
5/5 = 1

This sequence is decreasing powers of 5.

5^5 = 3125
5^4 = 625
5^3 = 125
5^2 = 25
5^1 = 5
5^0 = 1

So, it seems intuitive from this example that 5^0 must be 1.

Here's the second explanation, and it uses a well-known property of exponents.

If the bases are the same in a ratio of powers, then we can subtract the lower exponent from the upper exponent.

5^n/5^m = 5^(n - m)

For example, if the upper exponent n = 7 and the lower exponent m = 3, then the property shows:

5^7/5^3 = 5^(7 - 3) = 5^4

Let's try this again, but first realize the following.

5^4/5^4 = 1

We know that the value of this ratio must be 1 because the numerator and denominator are equal.

5^4/5^4 = 5^(4 - 4) = 5^0 = 1

 

[Deleted member 4993]

I can explain
5^2 = 5x5 = 25
5^3 = 5x5x5 = 125

But how to explain 5^0 = 1?

Thanks

Do you understand:

\(\displaystyle \frac{a^m}{a^n} \, = \, a^{m-n}\)

when m = n, the answer becomes inevitable.

 

[Denis]

when m = n, the answer becomes inevitable.

NO! The answer becomes 1 :idea:

However, if you first stated: "let inevitable = 1", then you'd be correct!

 

[Deleted member 4993]

when m = n, the answer becomes inevitable.

NO! The answer becomes 1 :idea:

However, if you first stated: "let inevitable = 1", then you'd be correct!

I said - I don't know english.....

 

[brucejin]

Thanks you so much.
I found you short explanation "Because we define it to be so." is easier to remember.