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Find the square root of the following number correct to 2 decimal places: 12.

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Hint- To find the square root of a number by factorization, first, find the prime factors of the number and make a group of triplets of the same numbers from the prime factors and then find their products. For example, the prime factor of \[\left( c \right) = a \times a \times b \times b \times a \times b = \underline {\left[ {a \times a} \right]} \times \underline {\left[ {b \times b} \right]} \].In this question, the square root of 12 is to be determined and should be approximated up to 2 decimal places only. So, first, we will factorize the integer 12 and then put the values of the square root which are not in pairs.


Complete step by step solution:

Let’s find the square root of 12 using the factorization method first we will factorize the given numbers only by the prime numbers.
\[
   2\underline {\left| {12} \right.} \\
   2\underline {\left| 6 \right.} \\
   3\underline {\left| 3 \right.} \\
   1 \\
 \]
Hence we can write \[\left( {12} \right) = 2 \times 2 \times 3 \times 1\]
Now make pair group of the factors of 12 as:
\[
  \left( {12} \right) = \underline {\left[ {2 \times 2} \right]} \times 3 \\
  \sqrt {12} = 2\sqrt 3 \\
 \]
Hence the square root of 12 is $ 2\sqrt 3 $.

Now, we know that the square root of 3 in decimal value is 1.732.
So, substitute $ \sqrt 3 = 1.732 $ in the equation $ \sqrt {12} = 2\sqrt 3 $ to determine the square root of the integer 12 up to 2 decimal places as:

$
  \sqrt {12} = 2\sqrt 3 \\
   = 2 \times 1.732 \\
   = 3.464 \\
   \approx 3.45 \\
 $
Hence, the square root of 12 approximated up to 2 decimal places is 3.45.

Note: Always find the prime factor of the number and group them into the pairs of similar factors. The square root of a number can either be found by using the estimation method or by factorization method. But the best and easy method of finding square roots is the factorization method as this has fewer calculations and saves time as well.

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