**.**

*N*^{0}= 1The 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, 2^{4} * 2 = 2^{4 + 1} = 2^{5}

and 2^{5} * 2 = 2^{5 + 1} = 2^{6}

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

Thus, 2^{3} / 2 = 2^{3 - 1} = 2^{2}

and 2^{2} / 2 = 2^{2 - 1} = 2^{1}

and, finally, 2^{1} / 2 = 2^{1 - 1} = 2^{0}.

So what is ** 2^{0}**?

If we evaluate the other expressions, we have the following:

8 / 2 = 4; 4 / 2 = 2; 2 / 2 = 1. So 2

^{0}= 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__

__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 ** (2^{3})^{4} = 2^{(3*4)} = 2^{12} = 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 N^{2}, I will get N, because N * N = N^{2}.

If I take the positive square root of N^{4}, I will get N^{2}, because N^{2} * N^{2} = N^{4}, 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__

__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...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*

(16

((16

(((16

((((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.

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