Exercise 31 Page 624

Let's begin with recalling that the geometric mean of two positive numbers a and b is the number x that satisfies the following equation. x=sqrt(a* b) In our exercise, we are given that the geometric mean of a number and 4 times the number is 22. Let m represent a number. This means that the second number will be 4m. Let's substitute these numbers into the above equation. 22=sqrt(m* 4m) Now we will solve the equation for m.

22=sqrt(m*4m)

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Solve for m

Multiply

Multiply

22=sqrt(4m^2)

RaiseEqn

LHS^2=RHS^2

22^2=(sqrt(4m^2))^2

PowSqrt

( sqrt(a) )^2 = a

22^2=4m^2

CalcPow

Calculate power

484=4m^2

RearrangeEqn

Rearrange equation

4m^2=484

DivEqn

.LHS /4.=.RHS /4.

m^2=121

Our next step will be to take a square root of both sides of the equation. Notice that, since m is a positive number, we will only consider a positive case when taking a square root of m.

m^2=121

SqrtEqn

sqrt(LHS)=sqrt(RHS)

sqrt(m^2)=sqrt(121)

SqrtPowToNumber

sqrt(a^2)=a

m=sqrt(121)

CalcRoot

Calculate root

m=11

The number is 11.