Fractional Mathematical Radicals
So, you think you've conquered exponent rules, mastered negative exponents, and even shown radicals who's boss? Then why not give fractional exponents a try?
Question: When do we use fractional exponents?
Answer: We can write any radical expression using a fractional exponent. A simple equation like 
is literally the same as 361/2 = 6. Seriously. They mean exactly the same thing. This also means that 
and 
.
While this may be a breakthrough of epic proportions, there is much more. Not only can we write radicals using fractional exponents, we can also combine indices with an exponent.
FYI: "Indices" is plural for index.
So 
is the same as x3/4, and 
is the same as (z – 1)57/71. Just stick the index of the radical in the denominator, with the original exponent in the numerator. This is useful because now we can apply all our exponent rules to radicals, too. Scizzore.
Sample Problem
Simplify the following: 
.
First thing we're going to do is rewrite this equation using only exponents. Fractions, here we come.
x2x4/3
Next, we know we've got to add exponents. This means we'll need to make sure 2 is rewritten as 6/3 before we add things up for our final answer.
x2x4/3 = x6/3x4/3 = x10/3
Annnnd, we're done. For now.
Sample Problem
Simplify the following: 
.
Moving left to right, let's get our Shmoop on. The first term is easy.
The next term can't readily be simplified, unless we'd like to write it using a fractional exponent.
The question now is: Are we able to add 3 and 61/3? Whenever we run into a problem like this, our advice is to always compare it to another, more familiar situation. For instance, can x + y2 be simplified? Of course not. Problem solved.
Sample Problem
Simplify the following: 
.
It's back to the grind. This one forces us to move term by term once again. The whole number in the first term isn't too tough. However, the exponent is going to require some fractionalization. Yes, we did just make up a word.
2(x3)1/6 – (2x)2 + 4x1/2 = 2x3/6 – (2x)2 + 4x1/2
Simplifying the 3⁄6 exponent to ½ while simplifying that middle term gives us:
2x3/6 – (2x)2 + 4x1/2 = 2x1/2 – 4x2 + 4x1/2
What's nice here is that we can now add the like terms. Be careful; in this case, like terms are only those whose x factors have the same exponents.
2x1/2 – 4x2 + 4x1/2 = 6x1/2 – 4x2
That's as simple as we can make it.
Sample Problem
Simplify the following using positive exponents: 
.
Let's start at the beginning. For this problem, that means we'll start by simplifying things inside of that square root.
Next, we can take the square root of the numerator and denominator separately. Notice how the denominator will use our fractional exponent rules.
Next, we can multiply. Remember, it's really like the 2x is over an invisible 1. This makes our multiplication no different than regular old fraction multiplication.
Last but not least, we need to take care of the x's we have in the numerator and denominator. This is just a classic case of negative exponents. No sweat.
Of course, we're going to great lengths to show you every single step. If your Shmoopability is on the fast track, feel free to skip a step here or there. Eventually, it'll be no big deal if your actual work for this problem looks like:
