Algebra 1 ~ Chapter 11.1

Simplifying Radical Expressions

List of Common Perfect Squares

1 36 121

4 49 144

9 64 169

16 81 196

25 100 225

An expression that contains a radical sign (√ ) is a radical expression. There are many types of radical expressions (such as square roots, cube roots, fourth roots, and so on), but in this chapter, you will study radical expressions that contain only square roots.

Examples of radical expressions:

The expression under a radical sign is the radicand. A radicand may contain numbers, variables, or both. It may contain one term or more than one term.

Simplest Form of a Square-Root Expression – An expression containing square root is in simplest form when…

the radicand has no perfect square factors other than 1

the radicand has no fractions

there are no square roots in any denominator.

Remember that, indicates a nonnegative square root. When you simplify a square-root expression containing variables, you must be sure your answer is not negative. For example, you might think that but this is incorrect because you do not know if x is positive or negative.

In both cases This is the correct simplification of

If x = 3, then In this case,

If x = –3, then In this case,

Example 1: Using the Product Property of Square Roots, simplify each expression

Factor the radicand using perfect squares.

Product Property of Square Roots

Simplify.

a.)

When factoring the radicand, use factors that are perfect squares.

Helpful Hint

Example 2 – Simplify each radical expression

a.) √12 b.) √90

c.) √36 d.) √75

e.) √147 f.) √52

Multiply Square Roots

Example 3 - Simplify

a.) √3 · √15 b.) √2 · √24

c.) 2√3 · 3√3 d.) 4√5 · 2√6

Simplifying Square Roots with Variables

When finding the principal square root of an expression containing variables, be sure that the result is not negative.

Example 4A - Simplify

Product Property of Square Roots

Product Property of Square Roots

Where x is nonnegative

Example 4B – Simplify

√40x4y5z3

Example 5 - Simplify each radical expression.

a. b.

Example 6 - All variables represent nonnegative numbers.

A. B. C.

Example 6 - Simplify. All variables represent nonnegative numbers.D.

Caution!In the expression and 5 are not

common factors. is completely

simplified.

Rationalizing the Denominator

A fraction containing radicals is in the simplest form if no prime factors appear under the radical sign with an exponent greater than 1 and if NO RADICALS ARE LEFT IN THE DENOMINATOR.

Rationalizing the denominator of a radical expression is a method used to eliminate radicals from the denominator of a fraction.

Example 7 – Rationalizing the Denominator

3

10

Simplify each radical expression.

A.) B.)

6

2

Conjugates

*Binomials in the form p√q + r√s and p√q - r√s are called conjugates.

*Conjugates are useful when simplifying radical expressions because if p, q, r and s are rational numbers, their product is always a rational number with no radicals.

*For example, (3 + √2)(3 - √2) = (3)2 – (√2)2

= 9 – 2 or 7

Example 8 - Using Conjugates to Rationalize a Denominator

36

2

Simplify each radical expression

A.)

B.)

25

3