# Powers.Why do i get the result 1 when raise a number to 0 power ?

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To understand why the result of any number raised to the power 0 is 1, consider the property of exponents: a^(x - y) = a^x/a^y

For any number N, if x = y, the relation shown earlier gives:

N^(x - y) = N^x/N^y

As x = y

N^0 = N^x/N^x

If any number is divided by itself, the result is 1

N^0 = 1

This is the reason why the result of any number raised to the power 0 is 1.

To respond to this question, we'll create the sequence of power of 2.

We'll raise 2 to the second, third , fourth powers.

2^2 = 2*2 = 4

2^3 = 2*2*2 = 4*2 = 8 = 2*2^2

2^4 = 2*2*2*2 = 8*2 = 16 = 2*2^3

As we can see, we'll get a geometric sequence whose common ratio is r = 2.

Now, we'll raise 2 to negative powers:-2,-3,-4

2^-2 = 1/2^2 = 1/4

2^-3 = 1/2^3 = 1/8

2^-4 = 1/2^4 = 1/16

We'll write the geometric sequence:

..., 1/2^4, 1/2^3, 1/2^2, 1/2, 2^0, 2^1, 2^2, 2^3, 2^4, .....

We notice that if we want to get any next term of the sequence, w'll have to multiply the actual term by 2:

(1/2^4)*2 = 1/2^3

(1/2^3)*2 = 1/2^2

(1/2^2)*2 = 1/2^1

(1/2^1)*2 = 1

1*2 = 2^1

2^1*2 = 2^2

......................

We notice that for 2^0, we'll get 1.

We can create such sequences, taking any number instead of 2 and we'll get for x^0 the value 1, all the time.