Let's break down and solve the expression: [tex]\(6 \sqrt{20} + 2 \sqrt{240}\)[/tex].
1. Simplify the square roots:
- For [tex]\(\sqrt{20}\)[/tex]:
[tex]\[
\sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2 \sqrt{5}
\][/tex]
- For [tex]\(\sqrt{240}\)[/tex]:
[tex]\[
\sqrt{240} = \sqrt{16 \cdot 15} = \sqrt{16} \cdot \sqrt{15} = 4 \sqrt{15}
\][/tex]
2. Substitute the simplified roots back into the expression:
- Original expression:
[tex]\[
6 \sqrt{20} + 2 \sqrt{240}
\][/tex]
- Substitute [tex]\(\sqrt{20}\)[/tex] and [tex]\(\sqrt{240}\)[/tex]:
[tex]\[
6 \sqrt{20} + 2 \sqrt{240} = 6 \cdot (2 \sqrt{5}) + 2 \cdot (4 \sqrt{15})
\][/tex]
3. Simplify each term:
- For [tex]\(6 \cdot 2 \sqrt{5}\)[/tex]:
[tex]\[
6 \cdot 2 \sqrt{5} = 12 \sqrt{5}
\][/tex]
- For [tex]\(2 \cdot 4 \sqrt{15}\)[/tex]:
[tex]\[
2 \cdot 4 \sqrt{15} = 8 \sqrt{15}
\][/tex]
4. Combine the terms:
- Combine [tex]\(12 \sqrt{5}\)[/tex] and [tex]\(8 \sqrt{15}\)[/tex]:
[tex]\[
12 \sqrt{5} + 8 \sqrt{15}
\][/tex]
5. Numerically approximate the result:
- Approximate [tex]\(\sqrt{5}\)[/tex] and [tex]\(\sqrt{15}\)[/tex]:
[tex]\[
\sqrt{5} \approx 2.236
\][/tex]
[tex]\[
\sqrt{15} \approx 3.873
\][/tex]
- Calculate each term:
[tex]\[
12 \sqrt{5} \approx 12 \cdot 2.236 = 26.832
\][/tex]
[tex]\[
8 \sqrt{15} \approx 8 \cdot 3.873 = 30.984
\][/tex]
- Combine the values to obtain the final result:
[tex]\[
26.832 + 30.984 = 57.816
\][/tex]
Thus, the numerical approximation for the expression [tex]\(6 \sqrt{20} + 2 \sqrt{240}\)[/tex] is approximately [tex]\(57.816\)[/tex].
