Radical fractions aren't little rebellious fractions that stay out late, drinking and smoking pot. Instead, they're fractions that include radicals – usually square roots when you're first introduced to the concept, but later on your might also encounter cube roots, fourth roots and the like, all of which are called radicals too. Depending on exactly what your teacher is asking you to do, there are two ways of simplifying radical fractions: Either factor the radical out entirely, simplify it, or "rationalize" the fraction, which means you eliminate the radical from the denominator but may still have a radical in the numerator.
Canceling Radical Expressions From a Fraction
Consider your first option, factoring the radical out of the fraction. There are actually two ways of doing this. If the same radical exists in all terms in both the top and bottom of the fraction, you can simply factor out and cancel the radical expression. For example, if you have:
(2√3) / (3√3_)_
You can factor out both the radicals, because they're present in every term in the numerator and denominator. That leaves you with:
√3/√3 × 2/3
And because any fraction with the exact same non-zero values in numerator and denominator is equal to one, you can rewrite this as:
1 × 2/3
Or simply 2/3.
Simplifying the Radical Expression
Sometimes you'll be faced with a radical expression that doesn't have a concise answer, like √3 from the previous example. In that case you'll usually preserve the radical term just as it is, using basic operations like factoring or canceling to either remove it or isolate it. But sometimes there's an obvious answer. Consider the following fraction:
In this case, if you know your square roots, you can see that both radicals actually represent familiar integers. The square root of 4 is 2, and the square root of 9 is 3. So if you see familiar square roots, you can just rewrite the fraction with them in their simplified, integer form. In this case, you'd have:
This also works with cube roots and other radicals. For example, the cube root of 8 is 2 and the cube root of 125 is 5. So if you encountered:
(3√8) / (3√125)
You would, with a little practice, be able to see right away that it simplifies to the much simpler and easier to handle:
Rationalizing the Denominator
Often, teachers will let you keep radical expressions in the numerator of your fraction; but, just like the number zero, radicals cause problems when they turn up in the denominator or bottom number of the fraction. So, the last way you may be asked to simplify radical fractions is an operation called rationalizing them, which just means getting the radical out of the denominator. Often, that means the radical expression turns up in the numerator instead.
Consider the fraction
You can't easily simplify _√_5 to an integer, and even if you factor it out, you're still left with a fraction that has a radical in the denominator, as follows:
1/_√_5 × 4/1
So neither of the methods already discussed will work. But if you remember the properties of fractions, a fraction with any non-zero number on both top and bottom equals 1. So you could write:
√_5/√_5 = 1
And because you can multiply 1 times anything else without changing the value of that other thing, you can also write the following without actually changing the value of the fraction:
√_5/√5 × 4/√_5
Once you multiply across, something special happens. The numerator becomes 4_√_5, which is acceptable because your goal was simply to get the radical out of the denominator. If it shows up in the numerator, you can deal with it.
Meanwhile, the denominator becomes √_5 × √5 or (√_5)2. And because a square root and a square cancel each other out, that simplifies to simply 5. So your fraction is now:
4_√_5/5, which is considered a rational fraction because there is no radical in the denominator.