# Radical Expressions and Equations

**Study Strategy –Note Taking**

•Choosing a Seat

•Note-Taking Speed

•Things to Include

•Being an Active Learner

•Formatting Your Notes

•Rewriting Your Notes

•Write Down Each Step

**Section 8.1–Square Roots and Radical Notation**

**Concept –Radical Expressions, Square Roots
**

The principal square root of b, denoted ,

is the positive number a such that a

^{2}=b

The expression is called a radical

expression.

The sign is called a radical sign.

The expression contained inside the

radical sign is called the radicand.

**Example –Radical Expressions, Square Roots**

Simplify:

because 7^{2}
=49

**Example –Radical Expressions, Square Roots**

Simplify, assume x is nonnegative:.

because

**Concept –Product Property of Radicals**

For any index n>1and any nonnegative

numbers a and b,

**Example –Product Property of Radicals**

Simplify:

**Example –Product Property of Radicals**

Simplify:

**Concept –Quotient Property of Radicals**

For any index n>1and any nonnegative

number a and positive number b ,

**Example –Quotient Property of Radicals**

Simplify:

**Section 8.2 -Simplifying, Adding, & Subtracting Radical
Expressions**

**Concept –Simplifying Radicals
**

1. Completely factor the radicand.

2. Rewrite each factor as a product of two

factors. The exponent for the first factor should

be the largest multiple of the radical’s index that

is less than or equal to the factor’s original

exponent.

3. Use the product property to remove factors

from the radicand.

**Example –Simplifying Radicals
**

Simplify:

**Concept –Like Radicals**

Two radical expressions are called like

radicals if they have the same index and

the same radicand.

**Example –Like Radicals**

**Concept –Adding and Subtracting Radical Expressions
**

1. Simplify each radical completely.

2. Combine like radicals by adding/subtracting

the factors in front of the radicals.

**Example –Adding and Subtracting Radical Expressions**

Subtract:

**Example –Adding and Subtracting Radical Expressions**

Add:

**Section 8.3 –Multiplying and Dividing Radical
Expressions**

**Concept –Multiplying Radical Expressions
**

If the index of each radical is the same,

multiply factors in front of the radical by

each other, and multiply the radicands by

each other. Simplify the resulting radical if

possible.

**Example –Multiplying Radical Expressions**

Multiply:

**Concept –Multiplying Radical Expressions
**

If one or more of the expressions contains at

least two terms, multiply by using the

distributive property.

**Example –Multiplying Radical Expressions**

Multiply:

**Example –Multiplying Radical Expressions
**

Multiply:

**Concept -Multiplying Radical Expressions
**

To square a radical expression containing two or more terms, multiply the expression by itself.

Example -Multiplying Radical Expressions

Multiply:

**Concept –Rationalizing a Denominator with Only
One Term Containing a Radical with an Index of n**

1. Multiply both the numerator and denominator

by a radical expression that results in the

radicand being a perfect nth power.

2. Simplify the radical in the denominator.

3. Simplify the fraction by dividing out factors

that are common to the numerator and

denominator, if possible.

**Example -Rationalizing a Denominator with Only
One Term Containing a Radical with an Index of n**

Rationalize the denominator: