Select the correct answer.
Which calculation correctly uses prime factorization to write [tex]$\sqrt{240}$[/tex] in simplest form?
[tex]$\sqrt{240}=\sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5}=4 \sqrt{15}$[/tex]
[tex]$\sqrt{240}=\sqrt{2 \cdot 2 \cdot 2 \cdot 3 \cdot 10}=2 \sqrt{60}$[/tex]
[tex]$\sqrt{240}=\sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3}=4 \sqrt{3}$[/tex]
[tex]$\sqrt{240}=\sqrt{2 \cdot 2 \cdot 2 \cdot 3 \cdot 5}=2 \sqrt{30}$[/tex]
Answer (1)
Find the prime factorization of 240, which is 240 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 .
Express 240 as 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 .
Simplify the square root by pairing factors: 2 4 ⋅ 3 ⋅ 5 = 2 2 3 ⋅ 5 = 4 15 .
The correct simplification is 4 15 .
Explanation
Problem Analysis We are given the problem of simplifying 240 using prime factorization and we need to choose the correct calculation from the given options. First, we need to find the prime factorization of 240.
Prime Factorization From the prime factorization, we have 240 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 . Therefore, 240 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 .
Simplifying the Square Root Now, we simplify the square root by taking out pairs of factors: 240 = 2 4 ⋅ 3 ⋅ 5 = 2 4 ⋅ 3 ⋅ 5 = 2 2 15 = 4 15 .
Identifying the Correct Calculation Comparing our result with the given calculations, we find that the correct calculation is 240 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 = 4 15 .
Examples
Understanding how to simplify square roots using prime factorization is useful in various real-life situations. For instance, when calculating the length of the diagonal of a rectangle or the hypotenuse of a right triangle, you often end up with a square root that needs simplification. Imagine you are designing a rectangular garden with an area of 240 square feet and you want to find the length of the diagonal to plan the layout. Simplifying 240 to 4 15 helps in easier calculations and better understanding of the dimensions involved.
