Exercise 34 Page 266

To simplify the given expression, we will use the FOIL method. Recall that the acronym F OI L stands for First, Outer, Inner, and Last. (2sqrt(3)-sqrt(5))(sqrt(10)+4sqrt(6)) ⇕ ( 2sqrt(3))( sqrt(10)) + ( 2sqrt(3))( 4sqrt(6)) + (-sqrt(5))(sqrt(10)) + ( -sqrt(5))( 4sqrt(6))Let's use perfect squares to simplify our radicals to see if we can create like terms.

(2sqrt(3))(sqrt(10))+ (2sqrt(3))(4sqrt(6)) + (-sqrt(5))(sqrt(10)) + (-sqrt(5))(4sqrt(6))

SplitIntoFactors

Split into factors

(2sqrt(3))(sqrt(10))+ (2sqrt(3))(4sqrt(3 * 2)) + (-sqrt(5))(sqrt(5 * 2)) + (-sqrt(5))(4sqrt(6))

SqrtProd

sqrt(a* b)=sqrt(a)*sqrt(b)

(2sqrt(3))(sqrt(10))+ (2sqrt(3))(4sqrt(3) * sqrt(2)) + (-sqrt(5))(sqrt(5) * sqrt(2)) + (-sqrt(5))(4sqrt(6))

AddNeg

a+(- b)=a-b

(2sqrt(3))(sqrt(10))+ (2sqrt(3))(4sqrt(3) * sqrt(2)) -sqrt(5) * sqrt(5) * sqrt(2) - sqrt(5) * 4sqrt(6)

CommutativePropMult

Commutative Property of Multiplication

2sqrt(3) * sqrt(10)+ 2 * 4 * sqrt(3) * sqrt(3) * sqrt(2) -sqrt(5) * sqrt(5) * sqrt(2) -4 * sqrt(5) *sqrt(6)

MultSqrt

sqrt(a)* sqrt(a)= a

2sqrt(3) * sqrt(10)+ 2 * 4 * 3 * sqrt(2) -5 * sqrt(2) -4 * sqrt(5) *sqrt(6)

ProdSqrt

sqrt(a)*sqrt(b)=sqrt(a* b)

2sqrt(3 * 10) + 2 * 4 * 3 * sqrt(2) -5 * sqrt(2) -4 * sqrt(5 * 6)

Multiply

Multiply

2sqrt(30) + 24 sqrt(2) -5sqrt(2) -4sqrt(30)

Now we will factor out common factors to perform addition and subtraction.

2sqrt(30) + 24 sqrt(2) -5sqrt(2) -4sqrt(30)

CommutativePropAdd

Commutative Property of Addition

24 sqrt(2) -5sqrt(2) +2sqrt(30)-4sqrt(30)

FactorOut

Factor out sqrt(2)

(24 -5)sqrt(2) +2sqrt(30)-4sqrt(30)

FactorOut

Factor out sqrt(30)

(24 -5)sqrt(2) +(2-4)sqrt(30)

SubTerms

Subtract terms

19sqrt(2)+(-2)sqrt(30)

AddNeg

a+(- b)=a-b

19sqrt(2) - 2sqrt(30)