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.

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.

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

Use the product property of square roots:

Simplify.