Exercise 40 Page 249

To rationalize the denominator, 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 2 as exponents.

sqrt(h^3)/sqrt(8)

WritePow

Write as a power

sqrt(h^3)/sqrt(2^3)

WriteSum

Write as a sum

sqrt(h^(2+1))/sqrt(2^(2+1))

SumInExponent

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

sqrt(h^2* h^1)/sqrt(2^2* 2^1)

ExpandFrac

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

sqrt(h^2* h^1)* sqrt(2^1)/sqrt(2^2* 2^1)* sqrt(2^1)

ProdSqrt

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

sqrt(h^2* h^1* 2^1)/sqrt(2^2* 2^1* 2^1)

MultPow

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

sqrt(h^2* h^1* 2^1)/sqrt(2^2* 2^2)

Now that we've 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 consider the index of the radicals to see how we should format our solution. sqrt(a^n)= a if n is odd |a| if n is even Because our radical has an even root and the variables have odd exponents, we do not need to use absolute value symbols to simplify our expression.

sqrt(h^2* h^1* 2^1)/sqrt(2^2* 2^2)

ExponentOne

a^1=a

sqrt(h^2* h* 2)/sqrt(2^2* 2^2)

SqrtProd

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

sqrt(h^2)sqrt(h* 2)/sqrt(2^2)sqrt(2^2)

SqrtPowToNumber

sqrt(a^2)=a

hsqrt(h* 2)/2(2)

Multiply

Multiply

hsqrt(2h)/4