Power Of 0 = 1 (Explain Please)

[Nick-30]

Homework Statement

For part of my homework i need to explain:
Why does any number with a power of zero have a value of 1?

Homework Equations

100^0=1
81^0=1
5^0=1

The Attempt at a Solution

 

[mgb_phys]

Look at the exponential sequence if you write down the positive and negative powers of a number
eg 3:
3^-3 = 1/27
3^-2 = 1/9
3^-1 = 1/3
3^0 = 1
3^1 = 3
3^2 = 9
3^3 = 27

What's the pattern between each value?

edit fixed formating

 

[Nick-30]

I really don't know. Could you please give me the answer.

 

[Borek]

Check what happens if you multiply 3*1/27, 3*1/9...

 

[Nick-30]

Yeah but how do i explain why does any number with a power of 0 have a value of 1

 

[Gokul43201]

1. Do you know how to expand n^{b+c}?

2. Now, if you set b+c=0, write b in terms of c, and repeat step 1, what do you get?

 

[mgb_phys]

Yeah but how do i explain why does any number with a power of 0 have a value of 1

Because if you divide "any number" by itself you get 1 and if you multiply "1/any number" by "any number". Just look at the above example with any other number than 3.

 

[joeyar]

You know the rules for adding/subtracting exponents?

(a^b)*(a^c) = a^(b+c)

and

(a^b)/(a^c) = a^(b-c).

What happens when b = c?

You get (a^b)/(a^b) = a^(b-b) = a^0. But if you look to the LHS, you will see we have a number divided by itself which is 1.

Incidentally, it is not true that 'any number' to the power of 0 = 1: 0^0 is undefined.

 

[Дьявол]

As others explained, a^b*a^c=a^b+c. Now try for example:
3^-1*3¹=3/3=1
Same if you write as:
3^-1+1=3⁰=1.

Do you know how to explain now?