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Multiplying and Dividing Radical Expressions

Radical Expressions are math operations having multiple terms.

Radical Multiplication

√2(√18 + √6) = √36 + √12 = √( 6 • 6) + √(2 • 2 • 3) = 6 + 2√3

The FOIL Method (First-Outer-Inner-Last):

(2 + √3) (4 - 3√3) =

F = 2 • 4 = +8

O = 2 •∙ (-3√3) = -6√3

I = √3  • 4 = 4√3

L = √3  • (-3√3) = -3 • 3 = -9

8 - 6√3 + 4√3 - 9 = -1- 2√3

Radical Multiplication Math Example:

x2 + 8x + 13 when x = -4 + √3

(First, substitute “x” with -4 + √3)

(-4 + √3) 2 + 8(-4 + √3) + 13 =

(Next, remove square from expression)

[(-4 + √ 3) (-4 + √3)] + (-32 + 8√3  ) + 13 =

[ 16 - 4√3 - 4√3 + 3 ] - 32 + 8√3 + 13 = 16 - 8√3 + 3 - 32 + 8√3 + 13 =

16 + 3 - 32 + 13 - 8√3 + 8√3 = 0

Radicals Division by Rationalizing Denominators

Radical denominators having two or more terms:

(4 + 2√5 ) / (5 - √5)

First, remove Radicals from denominator using the conjugate 5 + √5

(4 + 2√5) (5 + √5) ÷ (5 - √5) (5 + √5) =

(20 + 4√5 + 10√5 + 10) ÷ (25 + 5√5 - 5√5 - 5) =

30 + 14√5 ÷ 20 = (15 + 7√5) / 10

Radical denominators that divide all terms of numerator:

(√32 + √24 - 3√6) ÷ √2 =

First, divide all terms by denominator √2 to remove radical from denominator.

[(√32 + √24 - 3√6) / √2] ÷ [√2 / √2] =

(√16 + √12 - 3√3) / 1 =

4 + 2√3 - 3√3 =

4 - √3