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)`.