Solve the Expression: Finding the Value of 1/a^n When a≠0

This question is actually a proof of the law of exponents for negative exponents. We will prove it by using two other laws of exponents:

a. The zero exponent law, which states that raising any number to the power of 0 (except 0) will give the result 1:

$X^0=1$

b. The law of exponents for division between terms with identical bases:

$\frac{b^m}{b^n}=b^{m-n}$

Let's return to the problem whilst paying attention to two things. The first is that in the denominator of the fraction there is a term with base $a$. The second thing is that according to the zero exponent law mentioned above in a' we can always write the number 1 as any number (except 0) to the power of 0. Given that $a\neq0$ we can state that:

$1=a^0$

Let's apply this to the problem:

$\frac{1}{a^n}=\frac{a^0}{a^n}$

Now that we have terms with identical bases in the numerator and denominator of the fraction , we can apply the law of division between terms with identical bases mentioned in b' in the problem:

$\frac{a^0}{a^n}=a^{0-n}=a^{-n}$

Let's summarize the steps above as follows:

$\frac{1}{a^n}=\frac{a^0}{a^n}=a^{-n}$

In other words, we proved the law of exponents for negative exponents and furthermore we understood why the correct answer is answer c.