_{} for all real numbers

_{} if both *x* and *y*
are non-negative, and

_{} if both *x* and *y*
are non-negative, and *y* is not zero

**WARNING**:
Never cancel something inside a radical with something outside of it:

_{} **WRONG!** If you
did this you would be canceling a 3 with_{}, and they are certainly not the same number.

The general plan for reducing the radicand is to
remove any perfect powers. We are only considering square roots here, so what
we are looking for is any factor that is a perfect square. In the following
examples we will assume that *x* is positive.

**Example:**

_{}

In this case the 16 was recognized as a perfect square and removed from the radical, causing it to become its square root, 4.

**Example:**

_{}

Although *x*^{3} is not a perfect
square, it has a factor of *x*^{2}, which is the square of *x*.

**Example:**

_{}

Here the perfect square factor is *x*^{4},
which is the square of *x*^{2}.

**Example:**

_{}

In this example we could take out a 4 and a factor
of *x*^{2}, leaving behind a 2 and one factor of *x*.

· The basic idea is to factor out anything that is “square-rootable” and then go ahead and square root it.