Exercise 19 Page 267

To simplify the given radical expression we need to rationalize a fraction with a binomial denominator. To do this, we multiply the numerator and denominator of the fraction by the conjugate of the denominator. First, we will find the conjugate by changing the sign of the second term of the expression.

Binomial	Conjugate
a + b	a - b
a - b	a + b

In this case, the conjugate of the denominator is 1+sqrt(2).

3/1-sqrt(2)

ExpandFrac

a/b=a * (1+sqrt(2))/b * (1+sqrt(2))

3(1+sqrt(2))/(1-sqrt(2))(1+sqrt(2))

Distr

Distribute 3

3+3 sqrt(2)/(1-sqrt(2))(1+sqrt(2))

ExpandDiffSquares

(a+b)(a-b)=a^2-b^2

3+3 sqrt(2)/1^2-(sqrt(2))^2

BaseOne

1^a=1

3+3 sqrt(2)/1-(sqrt(2))^2

PowSqrt

( sqrt(a) )^2 = a

3+3 sqrt(2)/1-2

SubTerm

Subtract term

3+3 sqrt(2)/-1

MoveNegDenomToNum

Put minus sign in numerator

-(3+3 sqrt(2))/1

DivByOne

a/1=a

-(3+3 sqrt(2))

Distr

Distribute -1

-3-3 sqrt(2)