Fractional Exponents

By the end of this section, you should be able to solve the fol lowing problems .

1. Ex press in radical form and simplify

2. Simplify the expression . (As sume that all variables are positive ).

3. Simplify the expression. (Assume that all variables are positive).

4. Simplify the expression. (Assume that all variables are positive).

2 Concepts

In section 8.2 we said that the rules of exponents apply to radicals, now in
section 8.5 we learn exactly why that is so. Actually, any radical expression
can be rewritten as an exp onential expression . By definition we write,

For example,

When we have square roots , we have an invisible power of 1 and an
invisible root of two . actually means This becomes clear when we
rewrite one radical in exponent form, Often, we have to simplify radicals
when they are taken out of exponent form. The next example will show this

2.1 Example

Write as a radical in simplified form.

In the next three examples, we use rules of exponents to simplify as much
as possible, and then rationalize denominators when necessary

2.2 Examples
1. Simplify

2. Simplify

3. Rewrite the expression in simplest radical form.

3 Facts

1. The radical expression is equivalent to the exponential expression

2. Any radical expression can be rewritten using fractional exponents and
then simplified using the laws of exponents .

3. Once an expression has been simplified using the laws of exponents, it
can be rewritten as a radical and simplified further or rationalized.
4. In general, all radical and exponential expressions must be simplified
and/or rationalized when presented as an answer.