Rational approximation is a method that uses a sequence of rational numbers to get closer and closer to a given number to estimate the value of the number. The method requires that you investigate the size of the number by examining its value for increasingly smaller powers of 10 (i.e., tenths, hundredths, thousandths, and so on).

Since √84 is not a perfect square, you would use rational approximation to determine its decimal expansion.

Begin by determining which two integers the number would lie:

√84 is between the integers 9 and 10 because

9^2 <= (√84)^2 <= 10^2

81 <= 84 <= 100

Next, determine which interval of tenths the number belongs: 

√84 is between 9.1 and 9.2 because

9.1^2 <= (√84)^2 <= 9.2^2

82.81 <= 84 <= 84.64

Next, determine which interval of hundredths the number belongs:

√84 is between 9.16 and 9.17 because

9.16^2 <= (√84)^2 <= 9.17^2

83.9056 <= 84 <= 84.0889

A good estimate of the value of √84 is 9.17 because 84 is closer to 84.0889 than it is to 83.9056.

Notice that with each step we are getting closer and closer to the actual value, 84. This process can continue using intervals of thousandths, ten‐thousandths, and so on.   

√84 = 9.17 approximately.