Simplify: $\sqrt{324}$
Answer (1)
We start with the prime factorization of 324: 324 = 2 2 ⋅ 3 2 ⋅ 3 2 .
Apply the product property of square roots: 324 = 2 2 ⋅ 3 2 ⋅ 3 2 .
Simplify each square root: 2 2 = 2 and 3 2 = 3 .
Multiply the results to get the final answer: 2 ⋅ 3 ⋅ 3 = 18 .
Explanation
Understanding the Problem We are asked to simplify the square root of 324, which means we need to find a number that, when multiplied by itself, equals 324. We are given the prime factorization of 324 as 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 . We can rewrite this as 2 2 ⋅ 3 2 ⋅ 3 2 .
Applying Square Root Properties Using the properties of square roots, we can rewrite the expression as: 324 = 2 2 ⋅ 3 2 ⋅ 3 2 We can separate the square root of the product into the product of the square roots: 2 2 ⋅ 3 2 ⋅ 3 2 = 2 2 ⋅ 3 2 ⋅ 3 2
Simplifying Individual Square Roots Now, we simplify each square root: 2 2 = 2 3 2 = 3 3 2 = 3
Calculating the Final Result Finally, we multiply the simplified terms together: 2 ⋅ 3 ⋅ 3 = 18 Therefore, 324 = 18 .
Final Answer Thus, the simplified form of 324 is 18 .
Examples
Square roots are used in many real-world applications, such as calculating the distance between two points in a coordinate plane or determining the length of the side of a square given its area. For example, if you have a square garden with an area of 324 square feet, you can use the square root to find the length of each side. In this case, each side would be 324 = 18 feet long. This concept is also used in physics to calculate speeds and distances.
