Answer:
(a) 4 1/2
(b) -2 4/5
(c) See attached
Step-by-step explanation:
You want these square roots using the perfect squares approximation method, and their plot on a number line:
Approximation method
There are different versions of the perfect squares approximation method. The simpler one tends to over-estimate the root, while the slightly more complex one tends to under-estimate the root. We will show both methods.
The first step is to determine the largest perfect square less than the number whose root is of interest. It is helpful to be familiar with the perfect squares of small integers:
2² = 4
3² = 9
4² = 16
and so on.
Next, write the number of interest in terms of that square and the difference from it. For example, ...
5 = 2² +1
8 = 2² +4
n = r² +d
Finally, the perfect square method approximations will be ...
√n ≈ r +d/(2r) . . . . slightly high (better when d < r)
√n ≈ r + d/(2r+1) . . . . slightly low (better when d > r)
(a) 2√5
We know 5 is between the perfect squares 4 = 2² and 9 = 3², so using the first approximation above, we have ...
5 = 2² +1
√5 ≈ 2 +1/(2·2) = 2 1/4
The value we really want is 2√5, so that value is ...
2√5 ≈ 2(2 1/4)
2√5 ≈ 4 1/2
(b) -√8
We know 8 is between the perfect squares 4 = 2² and 9 = 3². Using the second approximation above, we have ...
8 = 2² +4
√8 ≈ 2 +4/(2·2 +1) = 2 4/5
The value we really want is -√8, so that value is ...
-√8 ≈ -2 4/5
(c) Number line
The above root approximations are plotted on the number line in the attachment.
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Additional comment
Using the above relation for n, r, d, we can write the square root of n as the continued fraction ...
[tex]\sqrt{n}=r+\cfrac{d}{2r+\cfrac{d}{2r+\cfrac{d}{\dots}}}[/tex]
Given an approximation of a root, R, a better approximation of the root, R', can be found as ...
[tex]R'=r+\dfrac{d}{r+R}[/tex]
For example, a better approximation of √8 will be ...
R' = 2 +4/(2 +2 4/5) = 2 5/6
This iteration method converges much more slowly than Newton's method, which computes R' as ...
R' = (R +n/R)/2
