Find the nearest integer to the square root of 600.

Hint: Square root of a number is a number when multiplied together produces the original number. The square root of a number is \[x = \sqrt x \] and when the result is multiplied by itself produces \[\sqrt x \times \sqrt x = x\] the original number. The square root of a number can be produced either by the estimation number or by the prime factorization method.
To find the square root of a number using the method of prime factorization first we will have to find the prime factor of the number and these numbers are grouped in the pairs which are the same and then their product is found.
E.g. Prime factor of \[\left( c \right) = a \times a \times a \times b \times b \times a \times c \times c\] the number which are grouped in pair as \[\left( c \right) = a \times a \times a \times b \times b \times a \times c \times c = \underline {a \times a} \times \underline {a \times a} \times \underline {b \times b} \times \underline {c \times c} = a \times a \times b \times c\]. In the need to find the nearest integer of the square root of 600. So, we need to first find the square root of 600.

Complete step by step answer:
Using the prime factorization method to find square root by finding their prime factors as:
\[
   2\underline {\left| {600} \right.} \\
   2\underline {\left| {300} \right.} \\
   2\underline {\left| {150} \right.} \\
   3\underline {\left| {75} \right.} \\
   5\underline {\left| {25} \right.} \\
   5 \\
 \]
Hence, we can write: \[\left( {600} \right) = 2 \times 2 \times 2 \times 3 \times 5 \times 5\]
Now make a pair of the same numbers of factors:
\[\left( {600} \right) = \underline {2 \times 2} \times 2 \times 3 \times \underline {5 \times 5} \]
We can see all the factors do not make a pair hence we can say that \[600\] is not a perfect square number.
No, we have to check the perfect squared number near \[600\] who has a square root number.
\[{\left( {24} \right)^2} = 24 \times 24 = 576\] And \[{\left( {25} \right)^2} = 25 \times 25 = 625\] are the perfect squared numbers between which \[600\] lies.
Now find the difference between them as:
\[
  600 - 576 = 24 \\
  625 - 600 = 25 \\
 \]
Hence, we observe the number nearest to \[600\] is \[576\] whose square root is \[\sqrt {576} = 24\], so the nearest integer to the square root of \[600\] is 24.

Note: If all prime factors of a number do not make a pair then the number is not a perfect square number and in the case of a perfect cube the factors must make triplets.