Blog: Zeroth Power and Fractional Exponents

I'm starting a new, old math book (report to come), and I came across something interesting in the first chapter about numbers: another way to look at why N⁰ = 1.

The rules for exponents are simple:

N^a * N^b = N^{(a + b)}
and N^a / N^b = N^{(a - b)}

In the case of multiplying repeatedly by the same number:

N^a * N = N^{(a + 1)}

So, say, 2⁴ * 2 = 2^{4 + 1} = 2⁵
and 2⁵ * 2 = 2^{5 + 1} = 2⁶

Likewise, when dividing, repeatedly, by the same factor, the exponent would be decremented:

Thus, 2³ / 2 = 2^{3 - 1} = 2²
and 2² / 2 = 2^{2 - 1} = 2¹
and, finally, 2¹ / 2 = 2^{1 - 1} = 2⁰.

So what is 2⁰?
If we evaluate the other expressions, we have the following:
8 / 2 = 4; 4 / 2 = 2; 2 / 2 = 1. So 2⁰ = 1.

Note that we could continue the progression into negative exponents if we keep dividing. This will lead to fractions. Perhaps on another day...

Fractional exponents

What does it mean to have a fraction as an exponent?

Keep in mind that negative exponents have nothing to do with negative numbers. They create fractions. So fractional exponents won't create fractions.

Let's review one more rule about exponents:

(N^a)^b = N^(ab)

So (2³)⁴ = 2^(3*4) = 2¹² = 4096.

What if there is an exponent of 2/3 or 3/5? First, consider that 2/3 = (2)(1/3) and 3/5 = (3)(1/5). The 2 and 3 still mean the second and third powers, but what about the unit fractions of 1/3 or 1/5?

If I take the positive square root, which I'll abbreviate SQRT(), of N², I will get N, because N * N = N².
If I take the positive square root of N⁴, I will get N², because N² * N² = N⁴, etc.

So SQRT(N^a) will give (N^(1/2)a).
However, our rule tells us that (N^(1/2)a) = (N^a)^1/2.
So (N^a)^1/2 is another way to write SQRT(N^a), and N^1/2 is another way to write SQRT(N).

Fractions and Zero

Consider the following progression:

SQRT(16) = 4
SQRT( SQRT(16)) = 2
SQRT( SQRT( SQRT(16))) = 1.414...
SQRT( SQRT( SQRT( SQRT(16)))) = 1.189...
SQRT( SQRT( SQRT( SQRT( SQRT(16))))) = 1.090...

As you repeatedly take the square root, the answer will get closer and closer to 1.
If we rewrite that using exponents, we get the following:

16^(1/2) = 4
(16^(1/2))^(1/2) = 16^(1/4) = 2
((16^(1/2))^(1/2))^(1/2) = 16^(1/8) = 1.414...
(((16^(1/2))^(1/2))^(1/2))^(1/2) = 16^(1/16) = 1.189...
((((16^(1/2))^(1/2))^(1/2))^(1/2))^(1/2) = 16^(1/32) = 1.090...

As the denominator gets larger, the fraction gets smaller. As the denominator goes toward infinity, the fraction goes toward zero. And the value on the right side of the equal sign goes toward 1.