Adding Exponents
To understand addition of exponents, let's start from a simple example.
Example. Suppose, we want to find $$${{2}}^{{3}}\cdot{{2}}^{{4}}$$$.
We already learned about positive integer exponets, so we can write, that $$${{2}}^{{3}}={2}\cdot{2}\cdot{2}$$$ and $$${{2}}^{{4}}={2}\cdot{2}\cdot{2}\cdot{2}$$$.
So, $$${\color{red}{{{{2}}^{{3}}}}}\cdot{\color{green}{{{{2}}^{{4}}}}}={\color{red}{{{2}\cdot{2}\cdot{2}}}}\cdot{\color{green}{{{2}\cdot{2}\cdot{2}\cdot{2}}}}={{2}}^{{7}}$$$.
Let's see what have we done. We counted number of 2's in $$${{2}}^{{3}}$$$, then counted number of 2's in $$${{2}}^{{4}}$$$. Since we multiplied, then we added number of 2's. Note, that $$${3}+{4}={7}$$$.
It appears, that this rule works not only for positive integer exponents, it works for any exponent.
Rule for adding exponents: $$$\color{purple}{a^m\cdot a^n=a^{m+n}}$$$.
Word of caution. It doesn't work, when bases are not equal.
For example, $$${{3}}^{{2}}\cdot{{4}}^{{5}}={3}\cdot{3}\cdot{5}\cdot{5}\cdot{5}\cdot{5}\cdot{5}$$$ which is neither $$${{3}}^{{7}}$$$ nor $$${{4}}^{{5}}$$$.
Word of caution. Above rule doesn't work for addition and subtraction.
For example, $$${{2}}^{{3}}+{{2}}^{{4}}\ne{{2}}^{{7}}$$$, because $$${{2}}^{{3}}+{{2}}^{{4}}={8}+{16}={24}$$$ and $$${{2}}^{{7}}={128}$$$. Clearly, $$${24}\ne{128}$$$.
Example 2. Find $$${{2}}^{{3}}\cdot{{2}}^{{-{5}}}$$$.
It doesn't matter, that exponent is negative.
Just proceed as always: $$${{2}}^{{3}}\cdot{{2}}^{{-{5}}}={{2}}^{{{3}+{\left(-{5}\right)}}}={{2}}^{{-{2}}}=\frac{{1}}{{{2}}^{{2}}}=\frac{{1}}{{4}}$$$.
Even when exponents are fractional, we use the same rule!
Example 3. Find $$${{3}}^{{\frac{{1}}{{4}}}}\cdot{{3}}^{{\frac{{2}}{{3}}}}$$$.
$$${{3}}^{{\frac{{1}}{{4}}}}\cdot{{3}}^{{\frac{{2}}{{3}}}}={{3}}^{{\frac{{1}}{{4}}+\frac{{2}}{{3}}}}={{3}}^{{\frac{{11}}{{12}}}}={\sqrt[{{12}}]{{{{3}}^{{11}}}}}$$$.
We can handle radicals, also, because radicals can be rewritten with the help of exponent.
Example 4. Rewrite, using positive exponent: $$${\sqrt[{{8}}]{{{3}}}}\cdot{\sqrt[{{7}}]{{\frac{{1}}{{{3}}^{{2}}}}}}$$$.
First we rewrite numbers, using exponents and then apply the rule:
$$${\sqrt[{{8}}]{{{3}}}}\cdot{\sqrt[{{7}}]{{\frac{{1}}{{{3}}^{{2}}}}}}={{3}}^{{\frac{{1}}{{8}}}}\cdot{\sqrt[{{7}}]{{{{3}}^{{-{2}}}}}}={{3}}^{{\frac{{1}}{{8}}}}\cdot{{3}}^{{-\frac{{2}}{{7}}}}={{3}}^{{\frac{{1}}{{8}}+{\left(-\frac{{2}}{{7}}\right)}}}={{3}}^{{-\frac{{9}}{{56}}}}=\frac{{1}}{{{{3}}^{{\frac{{9}}{{56}}}}}}$$$.
Now, it is time to exercise.
Exercise 1. Find $$${{3}}^{{2}}\cdot{{3}}^{{5}}$$$.
Answer: $$${{3}}^{{7}}={2187}$$$.
Exercise 2. Can we use rule for adding exponents to find $$${{2}}^{{5}}\cdot{{3}}^{{5}}$$$?
Answer: No, bases are not equal.
Exercise 3. Find $$${{4}}^{{\frac{{1}}{{3}}}}\cdot{{4}}^{{\frac{{2}}{{3}}}}$$$.
Answer: $$${4}$$$.
Exercise 4. Find $$${{3}}^{{2}}\cdot{{3}}^{{-\frac{{1}}{{5}}}}$$$.
Answer: $$${{3}}^{{\frac{{9}}{{5}}}}$$$.
Exercise 5. Find $$${\sqrt[{{7}}]{{\frac{{1}}{{27}}}}}\cdot{\sqrt[{{8}}]{{{9}}}}$$$.
Answer: $$${\sqrt[{{7}}]{{{{3}}^{{-{3}}}}}}\cdot{\sqrt[{{8}}]{{{{3}}^{{2}}}}}=\frac{{1}}{{{3}}^{{\frac{{5}}{{28}}}}}$$$.