To address the problem at hand, we need to find different ways to express [tex]\(\sqrt{240}\)[/tex] using its prime factorization.
First, let's start with the prime factorization of 240: [tex]\[ 240 = 2 \times 2 \times 2 \times 2 \times 3 \times 5 \][/tex]
Now, we want to express [tex]\(\sqrt{240}\)[/tex] in different forms. Here are four different ways to do so:
1. First way: [tex]\[
\sqrt{240} = \sqrt{2 \times 2 \times 2 \times 2 \times 3} = \sqrt{16 \times 3} = 4\sqrt{3}
\][/tex] The numerical value is approximately [tex]\(6.928203230275509\)[/tex].
2. Second way: [tex]\[
\sqrt{240} = \sqrt{2 \times 2 \times 2 \times 2 \times 3 \times 5} = \sqrt{16 \times 15} = 4\sqrt{15}
\][/tex] The numerical value is approximately [tex]\(15.491933384829668\)[/tex].
3. Third way: [tex]\[
\sqrt{240} = \sqrt{2 \times 2 \times 2 \times 3 \times 5} = \sqrt{4 \times 60} = 2\sqrt{30}
\][/tex] The numerical value is approximately [tex]\(10.954451150103322\)[/tex].
4. Fourth way: [tex]\[
\sqrt{240} = \sqrt{2 \times 2 \times 2 \times 3 \times 10} = \sqrt{4 \times 60} = 2\sqrt{60}
\][/tex] The numerical value is approximately [tex]\(15.491933384829668\)[/tex].
Each of these expressions represents [tex]\(\sqrt{240}\)[/tex] with different factors and their corresponding numerical values provide verification for the transformations.
