What is $\sqrt{200}$ in simplest form?
A. $2 \sqrt{10}$
B. $10 \sqrt{2}$
C. $100 \sqrt{2}$
D. $20 \sqrt{10}$
Answer (1)
Find the prime factorization of 200: 200 = 2 3 c d o t 5 2 .
Rewrite the square root: 200 = 2 3 c d o t 5 2 .
Simplify the square root by taking out perfect squares: 2 3 c d o t 5 2 = 10 2 .
The simplest form of 200 is 10 2 .
Explanation
Understanding the problem We are asked to simplify the square root of 200, which means we need to find the simplest form of 200 . The given options are A. 2 10 , B. 10 2 , C. 100 2 , D. 20 10 .
Finding the prime factorization To simplify 200 , we can find the prime factorization of 200.
Expressing 200 as a product of its prime factors The prime factorization of 200 is 200 = 2 3 ⋅ 5 2 = 2 ⋅ 2 2 ⋅ 5 2 .
Rewriting the square root Now we can rewrite the square root as 200 = 2 3 ⋅ 5 2 = 2 ⋅ 2 2 ⋅ 5 2 .
Simplifying the square root We simplify the square root by taking out the perfect squares: 2 ⋅ 2 2 ⋅ 5 2 = 2 ⋅ 2 2 ⋅ 5 2 = 2 ⋅ 2 ⋅ 5 = 10 2 .
Selecting the correct option Comparing the simplified expression 10 2 with the given options, we see that it matches option B. Therefore, the simplest form of 200 is 10 2 .
Examples
Simplifying radicals is useful in various fields, such as engineering and physics, where calculations often involve square roots. For instance, when calculating the length of the diagonal of a square with side length s , the diagonal is s 2 . If s = 10 , the diagonal is 10 2 . Simplifying radicals helps in obtaining more manageable and understandable expressions, making calculations easier and more intuitive. This skill is also fundamental in geometry when dealing with areas and volumes of shapes involving radicals.
