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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 a2 =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 72 =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: