Section 7.4 Operations with Radicals
Objective 1: Add and Subtract Radical Expressions.
Definition Like Radicals
Two radicals are like if they have the same index and the same radicand.
We add and subtract like radicals in the same way that we add and subtract like terms: reverse the distributive property to factor out the like radical and then simplify.
Sometimes it is necessary to simplify radicals first before adding or subtracting.
7.4.11 Add and subtract radical expressions.
7.4.17 Add. Assume all variables represent positive real numbers.
Objective 2: Multiply Radical Expressions
To multiply radical expressions, we follow the same approach as when multiplying polynomial expressions. We use the distributive property to multiply each term in the first expression by each term in the second. Then we simplify the resulting products and combine like terms and like radicals.
7.4.24 Multiply. If possible, simplify any radical expressions that appear in the product.
7.4.27 Multiply.

7.4.33 Multiply.
Objective 3: Rationalize Denominators of Radical Expressions
Rationalizing a Denominator with One Term
To rationalize a denominator with a single radical of index n , multiply the numerator and denominator by a radical of index n so that the radicand in the denominator is a perfect n th power.
7.4.39 Rationalize the denominator.
7.4.42 Rationalize the denominator.
Rationalizing a Denominator with Two Terms
To rationalize a denominator with two terms involving one or more square roots, multiply the numerator and denominator by the conjugate of the denominator.
7.4.47 Rationalize the denominator.
7.4.50 Rationalize the denominator.