- Topics At a Glance
**Exponents**- Negative Exponents
**Fractional Exponents**- Irrational Exponents
- Variables as Exponents
- Defining Polynomials
- Degrees of a Polynomial
- Multivariable Polynomials
- Degrees of Multivariable Polynomials
- Special Kinds of Polynomials
- Evaluating Polynomials
- Roots of a Polynomial
- Combining Polynomials
- Multiplying Polynomials
- Multiplication of a Monomial and a Polynomial
- Multiplication of Two Binomials
- Special Cases of Binomial Multiplication
- General Multiplication of Polynomials
- We'll Divide Polynomials Later!
- Factoring Polynomials
- The Greatest Common Factor
- Recognizing Products
- Trial and Error
- Factoring by Grouping
- Summary
- Introduction to Polynomial Equations
- Solving Polynomial Equations
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

Fractional exponents are abbreviations for taking the **roots** of a number. For example, *x*^{½} means , the **square root** of *x*. This makes sense because, by the rules of addition for exponents,

(*x*^{½})(*x*^{½}) = *x*^{(½+½)} = *x*^{1} = *x*

In the same way, *x*^{⅓} is the **cube root** of *x*. We need to multiply *x*^{⅓} by itself three times to get back to *x*. *x*^{¼} is the **fourth root** of *x*, *x*^{⅕} is the **fifth root** of *x*, and so on. What's that? You want one more? All right, all right. *x*^{⅙} is the **sixth root** of *x*, but that's all you're getting from us.

What about fractions that have a number other than 1 in the numerator? Can we ignore them and move on to easier problems? Unfortunately, no. That big gaping hole on your answer sheet will be glaring. What would *x*^{¾} mean? Well, if we multiply *x*^{¾} by itself 4 times, we get *x*^{3}. Therefore, *x*^{¾} is still a root; it's the fourth root of *x* cubed. By the rules of multiplying exponents, we can also write *x*^{¾} as (*x*^{3})^{¼}, or as (*x*^{¼})^{3}.

We could write it many other ways as well, but those other ways would be wrong, and if simplifying an exponent is wrong, we don't want to be right. Wait, what?

8^{⅔} can also be written as
**(8 ^{1/3})^{2}** . Since the cube root of 8 is 2, this simplifies to (2)

can be written as which simplifies to .

Exercise 1

Rewrite the following expression without fractional exponents, simplifying if possible:

100^{½}

Exercise 2

Rewrite the following expression without fractional exponents, simplifying if possible:

27^{4/3}

Exercise 3

Rewrite the following expression without fractional exponents, simplifying if possible: