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6.3 Binomial Radical Expressions
Like radicals: Are radical expressions that have the same index and the same radicand. To add or
subtract like radicals, focus on the coefficients and apply the operation needed. If the index and
radicand are different, the radicals cannot be combined.
Example 1: Adding and Subtracting Radical Expressions
1a) Add or subtract if possible. 5 ³√x - 3 ³√x
1b) 4√2 - 5√3
1c) 4√xy + 5√xy
Example 2: Simplifying Before Adding or Subtracting
 Simplify radicals before adding or subtracting so you can find all the like radicals.
2a) Simplify 6√18 + 4√8 - 3√72.
Step 1: Factor each radicand and simplify.
Step 2: Make sure to multiply the existing coefficient with whatever is taken out of the radicand.
Step 3: Combine the Coefficients.
2b) Simplify √50 + 3√32 - 5√18
II. Multiplying and Dividing Radical Expressions
 Multiply radical expressions that are in the form of binomials by using FOIL.
Example 3: Multiplying Binomial Radical Expressions
3a) (3 + 2√5) (2 + 4√5)
3b) (√2- √3)²
Conjugates: Are expressions, such as √a + √b and √a - √b, that differ only in the sign of the
second terms. If a and b are rational numbers, then the product of these conjugates is a rational
number.

.
Let a and b represent rational numbers.
(√a + √b)(√a - √b) = (√a )² - (√b)² (The Difference or Sum of Perfect Squares)
(√a )² - (√b)² = a - b
Example 4: Multiplying Conjugates
4a) (2 + √3) (2 - √3)
4b) (√5 + √2) (√5 - √2)
Example 5: Rationalizing Binomial Radical Denominators
 To rationalize a denominator of a fraction that is binomial radical, you must multiply the
denominator and numerator by the conjugate of the denominator.
6a) 3 + √5
1 - √5
6b) 5 - √21
√3 - √7
6c) 5 + 4√x
4
√x