# 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).