Exercise 28 Page 624

For any two positive numbers a and b, the geometric mean is the positive number x such that ax= xb. Since we know that x must be positive, the following equation is the definition of a geometric mean. x= sqrt(a b)We are asked to find the geometric mean of the given pair of numbers. 3sqrt(5)/4 and 5sqrt(5)/4 Let's substitute them into the equation and simplify the right-hand side to find the mean x.

x=sqrt(3sqrt(5)/4* 5sqrt(5)/4)

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Evaluate

MultFrac

Multiply fractions

x=sqrt(15 * sqrt(5) * sqrt(5)/16)

MultSqrt

sqrt(a)* sqrt(a)= a

x=sqrt(15 * 5/16)

Multiply

Multiply

x=sqrt(75/16)

SqrtQuot

sqrt(a/b)=sqrt(a)/sqrt(b)

x = sqrt(75)/sqrt(16)

SplitIntoFactors

Split into factors

x = sqrt(25 * 3)/sqrt(16)

SqrtProd

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

x = sqrt(25) * sqrt(3)/sqrt(16)

CalcRoot

Calculate root

x = 5sqrt(3)/4

Using a calculator, we can see that the obtained number is approximately 2.2.