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.
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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.
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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.
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The root is actually equal to the continued fraction ...
[tex]\sqrt{n}=\sqrt{s^2+r}=s+\cfrac{r}{2s+\cfrac{r}{2s+\dots}}[/tex]
