# Fractional (Rational) Exponents

Fractional exponent is a natural extension to the integer exponent.

We already know, that if b is positive integer, then

• $$a^b=\underbrace{a \cdot a \cdot a \cdot...\cdot a}_{b}$$$(see positive exponent) • $$a^{-b}=\frac{1}{a^b}=\frac{1}{\underbrace{a \cdot a \cdot a \cdot...\cdot a}_{b}}$$$ (see negative exponets)

But what if exponent is a fraction?

What will be the result of a^(m/n)?

We need nth root here:

Huge color(purple)(a^(m/n)=root(n)(a^m))

Such numbers are called radicals (rational exponents).

Now, let's go through examples.

Example 1. Rewrite using exponent: sqrt(3).

We can rewrite is root(2)(3^1).

Now, we clearly see, that sqrt(3)=3^(1/2).

Now, let's deal with negative exponents.

Example 2. Rewrite, using positive exponent: root(4)(1/27).

root(4)(1/27)=root(4)(1/3^3)=root(4)(3^(-3))=3^(-3/4)=1/(3^(3/4)).

So, root(4)(1/27)=1/(3^(3/4)).

Now, let's do inverse operation.

Example 3. Rewrite, using radicals: (2/5)^(3/7).

We just go in another direction: (2/5)^(3/7)=root(7)((2/5)^3)=root(7)(8/125).

Same applies to negative exponents.

Example 4. Rewrite, using radicals: (9/5)^(-7/8).

We have two ways here.

First is to get rid of minus on first step: (9/5)^(-7/8)=(5/9)^(7/8)=root(8)((5/9)^7).

Second way is to get rid of minus at last: (9/5)^(-7/8)=root(8)((9/5)^(-7))=root(8)((5/9)^7).

Now, exercise, to master this topic.

Exercise 1. Rewrite, using positive exponets: root(3)(5).

Answer: 5^(1/3).

Exercise 2. Rewrite, using positive exponets: root(4)((2/3)^(-3)).

Answer: (3/2)^(3/4).

Exercise 3. Rewrite, using radicals: 3^(2/7).

Answer: root(7)(9).

Exercise 4. Rewrite, using radicals: 7^(-1/2).

Answer: 1/sqrt(7).

Exercise 5. Rewrite, using radicals: (2/5)^(-3/7).

Answer: root(7)((5/2)^3)=root(7)(125/8).