Multiplication and Division of Roots
Suppose you were to see the equation
and you were asked to solve for x. What would you do? Well,
. And by the way, the radical sign always indicates the positive root, so 
never equals −2, by definition. This way, the result of the square-root operation is always just one number. So anyway, you can rewrite the equation as 3 + 2 = x, and you would know that x = 5. Because 4 is a perfect square, you were able to simply evaluate the root, and continue to solve the problem. But what if the equation were
and you were asked to find x? What would you do then? Neither 8 nor 2 is a perfect square, so you can’t easily find a value for either root.
It is important to realize that, on the GRE, sometimes you will be able to evaluate roots (when asked to take the square root of a perfect square or the cube root of a perfect cube), but other times it will be necessary to manipulate the roots. Up next is a discussion of the different ways that you are allowed to manipulate roots, followed by some examples of how these manipulations may help you arrive at a correct answer on GRE questions involving roots.
Go back to the previous question: If
what is x?
When two roots are multiplied by each other, you can do the multiplication within a single root. What that means is that you can rewrite
as
which equals
. And
equals 4, which means that x = 4.
This property also works for division.
If
,
what is x?
You can divide the numbers inside the square roots and put them inside one square root. So
becomes
, which becomes
And
equals 3, so x = 3.
Note that these rules apply if there are any number of roots being multiplied or divided. These rules can also be combined with each other. For instance,
becomes
The numbers inside can be combined, and ultimately you end up with
which equals 6.
