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

Example. Suppose, we want to find 2^3*2^4.

We already learned about positive integer exponets, so we can write, that 2^3=2*2*2 and 2^4=2*2*2*2.

So, color(red)(2^3)*color(green)(2^4)=color(red)(2*2*2)*color(green)(2*2*2*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: huge color(purple)(a^m*a^n=a^(m+n)).

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

For example, 3^2*4^5=3*3*5*5*5*5*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!=2^7, because 2^3+2^4=8+16=24 and 2^7=128. Clearly, 24!=128.

Example 2. Find 2^3*2^(-5).

It doesn't matter, that exponent is negative.

Just proceed as always: 2^3*2^(-5)=2^(3+(-5))=2^(-2)=1/2^2=1/4.

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

Example 3. Find 3^(1/4)*3^(2/3).

3^(1/4)*3^(2/3)=3^(1/4+2/3)=3^(11/12)=root(12)(3^11).

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

Example 4. Rewrite, using positive exponent: root(8)(3)*root(7)(1/3^2).

First we rewrite numbers, using exponents and then apply the rule:

root(8)(3)*root(7)(1/3^2)=3^(1/8)*root(7)(3^(-2))=3^(1/8)*3^(-2/7)=3^(1/8+(-2/7))=3^(-9/56)=1/(3^(9/56)).

Now, it is time to exercise.

Exercise 1. Find 3^2*3^5.

Answer: 3^7=2187.

Exercise 2. Can we use rule for adding exponents to find 2^5*3^5?

Answer: No, bases are not equal.

Exercise 3. Find 4^(1/3)*4^(2/3).

Answer: 4.

Exercise 4. Find 3^2*3^(-1/5).

Answer: 3^(9/5).

Exercise 5. Find root(7)(1/27)*root(8)(9).

Answer: root(7)(3^(-3))*root(8)(3^2)=1/3^(5/28).