Question

Estimate √53 between two numbers to the tenths place.

1. Identify perfect squares: 72 = 49 and 82 = 64
2. Estimate between two whole numbers: between 7 and 8
3. Determine the closer whole number: closer to 7 because 53 is closer to 49 than to 64
4. Find the squares of tenths: 7.12= 50.41
7.22= 51.84
7.32= 53.29
7.42= 54.76
√53 lies between
and
.

Answer 1

53 lies between 7.2² and 7.3²

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 ...

`√(n)=√(s^2+r)=s+\cfrac{r}{2s+\cfrac{r}{2s+\dots}}`