Subtracting Exponents

To understand subtraction of exponents, let's start from a simple example.

Example. Suppose, we want to find $\frac{{{{2}}^{{7}}}}{{{{2}}^{{4}}}}$.

We already learned about positive integer exponets, so we can write, that ${{2}}^{{7}}={2}\cdot{2}\cdot{2}\cdot{2}\cdot{2}\cdot{2}\cdot{2}$ and ${{2}}^{{4}}={2}\cdot{2}\cdot{2}\cdot{2}$.

So, $\frac{{{{2}}^{{7}}}}{{{{2}}^{{4}}}}=\frac{{{2}\cdot{2}\cdot{2}\cdot{\color{red}{{{2}\cdot{2}\cdot{2}\cdot{2}}}}}}{{{\color{red}{{{2}\cdot{2}\cdot{2}\cdot{2}}}}}}={2}\cdot{2}\cdot{2}={{2}}^{{3}}$.

Let's see what have we done. We counted number of 2's in ${{2}}^{{7}}$, then counted number of 2's in ${{2}}^{{4}}$. Since we divided, we canceled common terms. Note, that ${7}-{4}={3}$.

It appears, that this rule works not only for positive integer exponents, it works for any exponent.

Word of caution. It doesn't work, when bases are not equal.

For example, $\frac{{{{4}}^{{5}}}}{{{{3}}^{{2}}}}=\frac{{{4}\cdot{4}\cdot{4}\cdot{4}\cdot{4}}}{{{3}\cdot{3}}}$ which is neither ${{4}}^{{3}}$ nor ${{3}}^{{3}}$.

Word of caution. Above rule doesn't work for addition and subtraction.

For example, ${{2}}^{{7}}-{{2}}^{{4}}\ne{{2}}^{{3}}$, because ${{2}}^{{7}}-{{2}}^{{4}}={128}-{16}={112}$ and ${{2}}^{{3}}={8}$. Clearly, ${112}\ne{8}$.

Let's go through a couple of examples.

Even when exponents are fractional, we use the same rule!

We can handle radicals, also, because radicals can be rewritten with the help of exponent.

Finally, we can see now, why ${{a}}^{{0}}={1}$.

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

Now, it is time to exercise.