Just as multiple roots can be combined to create one root, you can also take one root and break it apart into multiple roots. You may be asking, why would you ever want to do that? Well, suppose a question asked: If
what is x? You would combine them, and say that x equals
. Unfortunately,
will never be a correct answer on the GRE. The reason is that
can be simplified, and correct answers on the GRE are presented in their simplest forms. So now the question becomes, how can you simplify
?
What if you were to rewrite
as
As mentioned, you could also break this apart into two separate roots that are multiplied together, namely
And you already know that
equals 2, so you could simplify to
And in fact, that is the simplified form of
, and could potentially appear as the correct answer to a question on the GRE. Just to recap, the progression of simplifying
was as follows:
Now the question becomes, how can you simplify any square root? What if you don’t notice that 12 equals 4 times 3, and 4 is a perfect square? Amazingly enough, the method for simplifying square roots will involve something you’re probably quite comfortable with at this point—prime factorizations.
Take a look at the prime factorization of 12. The prime factorization of 12 is 2 × 2 × 3. So
can be rewritten as
Recall the first roots rule—any root times itself will equal the number inside. If
can be rewritten as
you can take that one step further and say it is
And you know that
You can generalize from this example and say that when you take the prime factorization of a number inside a square root, any prime factor that you can pair off can effectively be brought out of the square root. Try another example to practice applying this concept. What is the simplified form of
? Start by taking the prime factorization of 360:
360 = 2 × 2 × 2 × 3 × 3 × 5
Again, you are looking for primes that you can pair off and ultimately remove from the square root. In this case, you have a pair of 2’s and a pair of 3’s, so you can separate them:
Notice that the prime factorization of 360 included three 2’s. Two 2’s could be paired off, but that still left one 2 without a partner, therefore
represents the prime factors that cannot be paired off. This expression can now be simplified to
, which is
.
You might have seen right away that 360 = 36 × 10, so
The advantage of the prime factor method is that it will always work, even when you don’t spot a shortcut.
Simplify the following roots.
