# Dividing Exponents

We already saw division of exponents two times:

Rule for dividing exponents: huge color(purple)(root(n)(a^m)=a^(m/n)).

Word of caution. It is not always possible to interchange exponent and nth root, i.e. root(n)(a^m)!=(root(n)(a))^m.

It is always possible for positive number, but not for negative.

Example.

root(4)((-5)^2)=root(4)(25)=root(4)(5^2)=5^(2/4)=5^(1/2)=sqrt(5),

but (root(4)(-5))^2 even doesn't exists, because root(4)(-5) doesn't exist.

Next couple of examples just show a couple of common problems.

Example 2. Rewrite, using positive exponents: root(5)(2^(-1/3)).

Just apply above rule: root(5)(2^(-1/3))=2^((-1/3)/5)=2^(-1/15).

Now, just rewrite using positive exponent: 2^(-1/15)=1/2^(1/15).

What if we have a couple of radicals?

Example 3. Simplify: root(4)(root(3)(1/25)).

We start form innermost number: root(4)(root(3)(5^(-2)))=root(4)(5^(-2/3))=5^((-2/3)/4)=5^(-1/6)=1/5^(1/6)=1/root(6)(5).

Let's see how interchanging works.

Example 4. Simplify: (root(3)((-2)^4))^6.

root(3)((-2)^4)^6=(root(3)(16))^6=16^(6/3)=16^2=256.

Now, it is time to exercise.

Exercise 1. Rewrite, using positive exponents: root(6)(2^3).

Answer: 2^(1/2)=sqrt(2).

Exercise 2. Find root(4)((-5)^2) and (root(4)(-5))^2.

Answer: root(4)(-5^2)=sqrt(5) and (root(4)(-5))^2 doesn't exist.

Exercise 3. Find root(5)((-2)^3).

Answer: (-2)^(3/5).

Exercise 4. Rewrite, using positive exponents: root(4)(root(5)(2^7).

Answer: 2^(7/20).

Exercise 5. Rewrite, using positive exponents: root(5)(root(3)(-2^5)).

Answer: -root(3)(2)=-2^(1/3).