Why is the 0 exponent equal 1?

Question

In using exponents WHY is 0 as an exponent to a number (5 to the 0 power) 1?

Answer 1

The zero exponent is equal to 1 to satisfy a certain case in manipulating bases with exponents. When multiplying or dividing the same base with exponents we add or subtract the exponents:

To find the value of (3^3)(3^2) [the short form of (3 X 3 X 3) X (3 X 3) or (27)(9)] we add exponents to get 3^(3+2) or 3^5 with a value of 243.

To find the value of (3^3)/(3^2)[the short form of (3X3X3)/(3X3) or 27/9] we subtract exponents to get 3^(3-2) or 3^1 with a value of 3.

Now let us consider the case in which the two exponents are the same:

To find the value of (3^3)/(3^3) [the short form of (3X3X3)/(3X3X3) or 27/27 we subtract exponents to get 3^(3-3) or 3^0 whose value must equal the value of 27/27, or 1. To make the equation true 3^0 must equal 1.

The general case is: (x^a)/(x^b)=x^(a-b). When b=a, by substitution, this becomes (x^a)/(x^a)=x^(a-a)=x^0. Now let’s consider it another way. On the left side of the equation we have a division of the same numerator and denominator, which has a value of 1 (anything divided by itself equals 1). On the right side we have x^0. To make the equation true x^0 must equal 1.

Answer 2

How about using the quotient rule to prove that x^0 = 1

x^3/x^3. Using the quotient rule we subtract the exponents, giving x^3-3 = x^0

Now if we give x a value, like 2, then 2^3/2^3 = 8/8 = 1

Hope this helps.

Answer 3

Think of "multiplication" as always including an extra "1" as a multiplier.  (Since 1 is the multiplication identity, the product is never changed by multiplying by 1).  When we multiply numbers, we can start with a 1 (identity) and then multiply it by the rest of the numbers, just like when we add or subtract numbers, we can start with 0 (identity), and then move to the right (addition) or left (subtraction) on the number line.

 

Now consider that an exponent (power) is merely the number of times you need to multiply the base (the five); and don't forget to also multiply by one (1).  So... 5 to the 3rd power would be 5 x 5 x 5 x 1 = 125, and 5 squared would be 5 x 5 x 1 = 25, etc.

Let's keep reducing the exponent (I'll write 5 to the n as 5^n):

5^3 = 5 x 5 x 5 x 1 = 125

5^2 = 5 x 5 x 1 = 25

5^1 = 5 x 1 = 5

5^0 = 1

If "5 to the 3rd" means multiply 1 using three fives (n=3), then 5 to the 0 means that I also want to multiply 1 using fives, but I want to use zero fives! (n=0).

Answer 4

Lets consider a simple example 1 = 5/5. Well, thats the same as 1 = (5^1)/(5^1). 

Now, move the 5 in the denominator to the numerator. That changes the sign of the exponent to -1: 1 = (5^1) x (5^ -1).

Now, you have a product of a base raised to different powers. Add the powers: 1 = 5^(1 - 1) = 5^0