To simplify the expression [tex]\(\sqrt{5}(8 + 3 \sqrt{6})\)[/tex], follow these steps:
1. Identify each term within the parentheses: The expression inside the parentheses is [tex]\(8 + 3 \sqrt{6}\)[/tex].
2. Distribute [tex]\(\sqrt{5}\)[/tex] to each term inside the parentheses. This means multiplying [tex]\(\sqrt{5}\)[/tex] by each term within the parentheses:
- For the first term, [tex]\(8\)[/tex]:
[tex]\[
\sqrt{5} \cdot 8 = 8\sqrt{5}
\][/tex]
- For the second term, [tex]\(3 \sqrt{6}\)[/tex]:
[tex]\[
\sqrt{5} \cdot 3 \sqrt{6}
\][/tex]
Multiply the constants and the radicals separately:
[tex]\[
\sqrt{5} \cdot 3 \sqrt{6} = 3 \cdot (\sqrt{5} \cdot \sqrt{6}) = 3 \sqrt{30}
\][/tex]
since [tex]\(\sqrt{5} \cdot \sqrt{6} = \sqrt{30}\)[/tex].
3. Combine the results from each distribution step:
[tex]\[
8 \sqrt{5} + 3 \sqrt{30}
\][/tex]
Thus, the simplified form of [tex]\(\sqrt{5}(8 + 3 \sqrt{6})\)[/tex] is:
[tex]\[
8 \sqrt{5} + 3 \sqrt{30}
\][/tex]
