Question

How do you simplify the square root of $200$?

Answer 1

Hint:
First we need to find the factors of the given number to calculate the square root. Factors are the number which completely divide the number without leaving any remainder behind. Square root of a $200$ can be calculated by prime factorization of the number. Prime factorization is defined as expressing a number as a product of prime numbers. The numbers which are divisible by $1$ and itself, i.e., it should only have two factors are called prime numbers.

Complete step by step solution:
Given the number to simplify is $\sqrt {200} $.
First we will find the square root of $200$ by finding its prime factors. Square root of a number can be simplified by prime factorization of the number.
Prime factorization involves expressing a number as a product of prime numbers.
The prime factorization of $200$is as follows,
 $\begin{gathered}
  2\left| {200} \right. \\
  2\left| {100} \right. \\
  2\left| {50} \right. \\
  5\left| {25} \right. \\
  5 \\
\end{gathered} $
Therefore, $200$ has $2$and $5$ as its prime factors.

Hence the factor of $(200) = 2 \times 2 \times 2 \times 5 \times 5$
Now we will pair the similar factors in a group of two,
Therefore,
$\sqrt {200} = \sqrt {2 \times 2 \times 2 \times 5 \times 5} $
$ \Rightarrow \sqrt {200} = \sqrt {\underline {2 \times 2} \times 2 \times \underline {5 \times 5} } $
$ \Rightarrow \sqrt {200} = 2 \times 5\sqrt 2 $
$ \Rightarrow \sqrt {200} = 10\sqrt 2 $

Hence, we get the square root of $200$ as $10\sqrt 2 $.

Note:
Square root of any number is calculated by prime factorization of the number. After prime factorization which involves expressing a number as a product of prime numbers (numbers which are divisible by $1$ and itself), we need to pair the similar factors in a group of two. Another way to simplify the square root of any number is using the long division method, which is quite complex, and thus, the chances of getting any error is high.