Certainly, let’s simplify the expression [tex]\(\sqrt{240}\)[/tex].
First, let's factorize 240 into products that include perfect squares:
1. Identify the prime factorization of 240:
[tex]\[
240 = 2^4 \times 3 \times 5
\][/tex]
2. From this factorization, we can group the factors to identify the largest perfect square:
[tex]\[
240 = (2^4) \times (3 \times 5) = 16 \times 15
\][/tex]
3. Now, using the property of square roots, [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], we can simplify:
[tex]\[
\sqrt{240} = \sqrt{16 \times 15} = \sqrt{16} \times \sqrt{15}
\][/tex]
4. Recognizing that [tex]\(\sqrt{16} = 4\)[/tex]:
[tex]\[
\sqrt{240} = 4 \times \sqrt{15}
\][/tex]
Thus, the simplified form of [tex]\(\sqrt{240}\)[/tex] is:
[tex]\[
\boxed{4 \sqrt{15}}
\][/tex]
Given options:
1. [tex]\(4 \sqrt{15}\)[/tex]
2. [tex]\(4 + \sqrt{15}\)[/tex]
3. [tex]\(16 \sqrt{15}\)[/tex]
4. [tex]\(16 + \sqrt{15}\)[/tex]
The correct option is [tex]\(4 \sqrt{15}\)[/tex].
Verifying numerically:
If we calculate [tex]\(4 \sqrt{15}\)[/tex]:
1. [tex]\(\sqrt{15} \approx 3.872\)[/tex]
2. [tex]\(4 \times 3.872 \approx 15.491933384829668\)[/tex]
So, [tex]\(\sqrt{240} \approx 15.491933384829668\)[/tex] which confirms our simplified form [tex]\(4 \sqrt{15}\)[/tex]. Therefore, the correct simplified form is:
[tex]\[
\boxed{4 \sqrt{15}}
\][/tex]
