the square root of141?

Question

approximate the expression involving square roots to three decimals

Answer 1

11.874

Answer 2

This one is too easy, since any calculator or computer with internet (as you have) can easily get it.

Perhaps the solution approach is more the point. Are you supposed to use a specific approach?

Example: Find √6 to 4 decimal places without a calculator

Since 2^2 = 4 and 3^2 = 9, we know that √6 is between 2 and 3. Let's just make a guess of it being 2.5. Squaring that we get 2.5^2 = 6.25. That's too high, so make the guess a little less. Let's try 2.4 next. To find approximation to four decimal places we need to do this till we have five decimal places, and then round the result.
Guess Square of guess High/low
2.4 5.76 Too low
2.45 6.0025 Too high but real close
2.449 5.997601 Too low
2.4495 6.00005025 Too high, so between 2.449 and 2.4495
2.4493 5.99907049 Too low
2.4494 5.99956036 Too low, so between 2.4494 and 2.4495
2.44945 5.9998053025 Too low, so between 2.44945 and 2.4495.

This is enough since we now know it would be rounded to 2.4495 (and not to 2.4494).

So, for 141, guesses with the appropriate square are:
11 121
12 144
11.5 132.25
11.7 136.89
11.8 139.24
11.9 141.61
11.89 141.3721
11.88 141.1344
11.87 140.8969
11.875 141.015625
11.874 140.991876

Answer 3

Let us define f(x)=√x.

Then, using the Taylor series expansion of f(x) at xₒ,
f(x)=f(xₒ) + f'(xₒ)(x-xₒ) + [f"(xₒ)(x-xₒ)^2]/2 + ...

let us set xₒ=144 for simplicity (√144=12).

f(x)=√x; f'(x)=1/(2√x); f"(x)=(-1/4)(x^(-3/2));...

Therefore,
f(141)=12 + 1/24*(141-144) + (-1/4)[144^(-3/2)](141-144)^2 + ...≈11.8736979≈11.874
which is a close enough approximation for our purposes (three decimals)

Answer 4

the square root of 141 is 11.87434209

Answer 5

√141 =11.874342087037916

Answer 6

11.87434209 or 11.874 rounded to three decimal places

Answer 7

you can just google

sqrt(141)

and get

sqrt(141) = 11.8743421, which is 11.874

Answer 8

√(141) =~ 11.874

Answer 9

11.874342087037 917234672917600412

Answer 10

11.874... or 11.874342087037917234672917600412...( it's non-terminating, non-repeating)