#
Section 8-1

#
Simplifying Radical Expressions

When you take the
square root
of a whole number, you either get another whole number (e.g.
) or you get an
irrational number
(e.g.
). Be careful when adding with irrational numbers:
an expression like
can't be simplified. It's already in simplest form. On the other hand, you can use the
distributive law
to group like radicals:

Irrational numbers can sometimes be simplified by factoring out a
perfect square
from the radicand (the part under the square root sign.) First, you need to know this important property:

##
PRODUCT PROPERTY OF SQUARE ROOTS

For all real numbers
*
a
*
and
*
b
*
,

That is, the square root of the product is the same as the product of the square roots.

There's an analogous quotient property:

For all real numbers
*
a
*
and
*
b
*
,
*
b
*
≠ 0:

##
SIMPLIFYING RADICALS

The idea here is to find a perfect square factor of the radicand, write the radicand as a product, and then use the product property to simplify.

Example:

Simplify.

9 is a perfect square, which is also a factor of 45.

Use the product property.

##
VARIABLE EXPRESSIONS UNDER THE RADICAL SIGN

When you have variables under the radical sign, see if you can factor out a square.

Example:

Simplify.

We can factor the radicand as the product of
*
a
*
and a squared expression.

Use the product property of square roots:

Simplify.