# Dividing Exponents

We already saw **division of exponents** two times:

- when discussed fractional exponents (`a^(m/n)=root(n)(a^m)`)
- when discussed multiplication of exponents (indeed, `a^(m/n)=a^(m*1/n)=(a^m)^(1/n)=root(n)(a^m)`).

**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)`.