Exercise 68 Page 420

To simplify the given expression, we can first rewrite the radical as a quotient of two radicals. sqrt(3x/2y)=sqrt(3x)/sqrt(2y) Now, we can rationalize the denominator of the quotient. We will multiply the numerator and denominator by a factor that will make the denominator a perfect square inside the square root. We will do this using the fact that we can multiply the radicands of radicals if they have the same index. If sqrt(a) and sqrt(b) are real numbers, then sqrt(a)* sqrt(b)= sqrt(ab). Let's start by finding the exponents necessary to create perfect squares in the denominator. Our goal is to have two of each factor.

sqrt(3x)/sqrt(2y)

WritePow

Write as a power

sqrt(3x)/sqrt(2^1y^1)

ExpandFrac

a/b=a * sqrt(2^1y^1)/b * sqrt(2^1y^1)

sqrt(3x)*sqrt(2^1y^1)/sqrt(2^1y^1)*sqrt(2^1y^1)

ProdSqrt

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

sqrt(3x*2^1y^1)/sqrt(2^1y^1*2^1y^1)

MultPow

a^m*a^n=a^(m+n)

sqrt(3x*2y)/sqrt(2^2y^2)

Now that we have found the factors that will make the radicand of the denominator perfect squares only, we can begin to simplify the quotient. While simplifying, we should remember that all variables are positive, so sqrt(a^n) = a.

sqrt(3x*2y)/sqrt(2^2y^2)

SplitIntoFactors

Split into factors

sqrt(2* x* y* 3)/sqrt(2^2y^2)

CommutativePropMult

Commutative Property of Multiplication

sqrt(2* 3* x* y)/sqrt(2^2y^2)

SqrtPowToNumber

sqrt(a^2)=a

sqrt(2* 3* x* y)/2y

Multiply

Multiply

sqrt(6xy)/2y