Subtracting Exponents

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

Example. Suppose, we want to find `(2^7)/(2^4)`.

We already learned about positive integer exponets, so we can write, that `2^7=2*2*2*2*2*2*2` and `2^4=2*2*2*2`.

So, `(2^7)/(2^4)=(2*2*2*color(red)(2*2*2*2))/(color(red)(2*2*2*2))=2*2*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.

Rule for subtracting 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, `(4^5)/(3^2)=(4*4*4*4*4)/(3*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!=2^3`, because `2^7-2^4=128-16=112` and `2^3=8`. Clearly, `112!=8`.

Let's go through a couple of examples.

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^(3+5)=2^8`.

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^(-5/12)=1/3^(5/12)`.

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^(23/56)`.

Finally, we can see now, why `a^0=1`.

Indeed, `a^0=a^(n-n)=(a^n)/(a^n)=1`.

Now, it is time to exercise.

Exercise 1. Find `(3^5)/(3^2)`.

Answer: `3^3=27`.

Exercise 2. Can we use rule for adding exponents to find `(5^5)/(3^5)`?

Answer: No, bases are not equal.

Exercise 3. Find `(4^(5/3))/(4^(2/3))`.

Answer: `4`.

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

Answer: `3^(2+1/5)=3^(11/5)=root(5)(3^11)`.

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

Answer: `root(7)(3^(-3))/root(8)(3^2)=1/3^(19/28)`.