Question

Estimate √53 between two numbers to the tenths place.

Answer 1

Answer:

  53 lies between 7.2² and 7.3²

Step-by-step explanation:

Estimating a root to the nearest tenth can be done a number of ways. The method shown here is to identify the tenths whose squares bracket the value of interest.

You have answered the questions of parts 1 to 3.

__

4.

You are given that ...

  7.2² = 51.84

  7.3² = 53.29

This means 53 lies between 7.2² and 7.3², so √53 lies between 7.2 and 7.3.

53 is closer to 7.3², so √53 will be closer to 7.3 than to 7.2.

7.3 is a good estimate of √53 to the tenths place.

_____

Additional comment

For an integer n that is the sum of a perfect square (s²) and a remainder (r), the square root is between ...

  s +r/(2s+1) < √n < s +r/(2s)

For n = 53 = 7² +4, this means ...

  7 +4/15 < √53 < 7 +4/14

  7.267 < √53 < 7.286

Either way, √53 ≈ 7.3.

__

The root is actually equal to the continued fraction ...

  $\sqrt{n}=\sqrt{s^2+r}=s+\cfrac{r}{2s+\cfrac{r}{2s+\dots}}$