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Multiplying and Dividing Radical Expressions

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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

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