Which of the following simplifies to $6 \sqrt{3}$?
A. $\sqrt{18}$
B. $\sqrt{54}$
C. $\sqrt{108}$
Answer (1)
18 simplifies to 3 2 . 54 simplifies to 3 6 . 108 simplifies to 6 3 . Therefore, the answer is 108 .
Explanation
Understanding the Problem We are given three options: 18 , 54 , and 108 . We want to determine which of these simplifies to 6 3 .
Simplifying 18 Let's simplify each of the given square roots.
18 = 9 ⋅ 2 = 9 ⋅ 2 = 3 2
Simplifying 54 54 = 9 ⋅ 6 = 9 ⋅ 6 = 3 6 = 3 2 ⋅ 3 = 3 2 3
Simplifying 108 108 = 36 ⋅ 3 = 36 ⋅ 3 = 6 3
Finding the Matching Expression Comparing the simplified forms to 6 3 , we see that 108 simplifies to 6 3 .
Examples
Simplifying radicals is a useful skill in various fields. For example, when calculating distances in geometry, you might encounter expressions like 108 . Simplifying this to 6 3 can make further calculations, such as finding areas or volumes, much easier. This skill is also useful in physics when dealing with energy or momentum calculations.
