Fractional Exponents

 

Learning Objective(s)

·         Simplify algebraic expressions with fractional exponents.

 

 

Square roots are most often written using a radical sign, like this:. But there is another way to represent the taking of a root. We can use fractional exponents instead of a radical.

 

Can’t imagine raising a number to a fractional power? They may be hard to get used to, but fractional exponents can actually help simplify some problems. Let’s see how these fractional exponents we call rational radicals work.

 

 

Radicals and exponents are inverse operations. It may be surprising, then, to learn that a radical can be expressed as an exponential number. The table below shows some examples of common square roots written as radicals, fractional exponents, and integers. Notice that the denominator of the fractional exponent is the number 2.

 

 

Let’s look at some more examples, but this time with cube roots. Remember, cubing a number raises it to the power of three. Notice that in these examples, the denominator of the fractional exponent is the number 3.

 

 

 

These examples help us model a relationship between radicals and fractional exponents: namely, that the nth root of a number can be written as either  or.

 

 

 

“The 5th root of the number 243” can be written as either:   A)   B)   C)   D)   Show/Hide Answer A) Correct. The 5th root of 243 can be expressed as a radical or as a fractional exponent with 5 in the denominator.   B) Incorrect. You have the exponent and the root switched. The 5th root of 243 can be expressed as a radical or as a fractional exponent with 5 in the denominator. The correct answer is A) .   C) Incorrect. You would use these expressions to write “the 243rd root of the number 5.” This will be a tiny number indeed! The correct answer is A) .   D) Incorrect. These two expressions are not equivalent.  is the 5th root of 243, while 2435 is 243 raised to the fifth power. The correct answer is A) .

 

 

 

All of the numerators for the fractional exponents in the examples above were 1. We can use exponents other than unit fractions to express roots, as shown below.  Notice any patterns within this table?

 

 

To write a radical as a fractional exponent, the power to which the base is raised becomes the numerator and the root becomes the denominator.

 

 

This makes sense for our unit fraction exponents as well. For example, the radical  can also be written as, since any number remains the same value if it is raised to the first power. We can now see where the numerator of 1 comes from in the equivalent form of .

 

 

 

Fractional exponents are not used very much, outside of advanced formulas in upper level math and science.  But occasionally they are useful in simplifying algebraic expressions.

 

Let’s explore some radical expressions now and see how. Here’s a radical expression that needs simplifying,.

 

One method of simplifying this expression is to factor and pull out groups of a³, as shown below:

 

 

 

 

We can also simplify this expression by thinking about the radical as a fractional exponent, and using the principle that any radical in the form  can be written as a fractional exponent in the form :

 

 

 

 

 

 

Note that fractional exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.

 

Both simplification methods gave us the same result, a². Depending on the context of the problem, it may be easier to use one method or the other, but for now, we’ll note that we were able to simplify this expression more quickly using fractional exponents than we were using the “pull-out” method.

 

Let’s try a more complicated expression, . Wow! This expression has two variables, a fraction, and a radical. That’s a little intimidating. We’ll take it step-by-step and see if using fractional exponents can help us simplify it.

 

We’ll start by simplifying the denominator because this is where the radical sign is located.

 

 

 

Well, that took a while, but we did it. We applied what we know about fractional exponents, negative exponents, and the rules of exponents to simplify the expression.

 

 

Simplify   A)   B)   C)   D)   Show/Hide Answer A) Incorrect.  . . The correct answer is C) .   B) Incorrect.  . . The correct answer is C) .   C) Correct. You can use fractional exponents and apply the rules of exponents to simplify  Next, you want to make sure there is no radical in the denominator, so you change the radical into a fractional exponent: . Now get the variable out of the denominator:. Add the exponents of a: . Convert the fractional exponent to a radical: . Finally, pull out the square root: = .    D) Incorrect. Try substituting a = 8 into the two expressions and see if they are equal. The correct answer is C) .

 

 

A radical can be expressed as a value with a fractional exponent by following the convention =. Rewriting radicals as fractional exponents can be useful in simplifying some radical expressions. When working with fractional exponents, remember that fractional exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.