Adding Exponents

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)`.