Exercise 53 Page 626

Let's remember the formulas to find the arithmetic and geometric mean of two numbers a and b. Arithmetic Mean: & a+b/2 Geometric Mean: & sqrt(ab) Notice that both, the arithmetic mean and the geometric mean, calculate a value that is between a and b. We know the arithmetic mean is the midpoint between a and b. In contrast, without knowing the numbers, we are not able to know where the geometric mean is.

To determine when the arithmetic and the geometric mean are equal, we must equate the two formulas. a+b/2 = sqrt(ab) Let's solve this equation.

a+b/2 = sqrt(ab)

MultEqn

LHS * 2=RHS* 2

a+b = 2sqrt(ab)

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Simplify

RaiseEqn

LHS^2=RHS^2

(a+b)^2 = (2sqrt(ab))^2

CalcPow

Calculate power

(a+b)^2 = 4ab

ExpandPosPerfectSquare

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

a^2 + 2ab + b^2 = 4ab

SubEqn

LHS-4ab=RHS-4ab

a^2 - 2ab + b^2 = 0

FacNegPerfectSquare

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

(a-b)^2 = 0

From the resulting equation, we conclude that a-b=0, and therefore, a=b. In consequence, the arithmetic and geometric mean are equal when the two numbers are equal (a=b).