Certainly! Let's simplify the expression step by step:
Given the expression:
[tex]\[
8 \sqrt{3}(\sqrt{3} + \sqrt{14})
\][/tex]
We will distribute [tex]\(8 \sqrt{3}\)[/tex] across the terms inside the parentheses. This means we will multiply [tex]\(8 \sqrt{3}\)[/tex] by [tex]\(\sqrt{3}\)[/tex] and by [tex]\(\sqrt{14}\)[/tex] separately:
First, consider the term [tex]\(8 \sqrt{3} \cdot \sqrt{3}\)[/tex]:
[tex]\[
8 \sqrt{3} \cdot \sqrt{3} = 8 (\sqrt{3} \cdot \sqrt{3}) = 8 \cdot 3 = 24
\][/tex]
Next, consider the term [tex]\(8 \sqrt{3} \cdot \sqrt{14}\)[/tex]:
[tex]\[
8 \sqrt{3} \cdot \sqrt{14} = 8 \cdot \sqrt{3 \cdot 14} = 8 \cdot \sqrt{42}
\][/tex]
Thus, the expression can be written as the sum of the two simplified terms:
[tex]\[
24 + 8 \sqrt{42}
\][/tex]
Evaluating [tex]\(8 \sqrt{42}\)[/tex] numerically:
[tex]\[
8 \sqrt{42} \approx 51.84592558726288
\][/tex]
Therefore, the simplified expression is:
[tex]\[
24 + 8 \sqrt{42}
\][/tex]
Numerically, the expression [tex]\(8 \sqrt{42}\)[/tex] approximately equals [tex]\(51.84592558726288\)[/tex], so the final numerical answer is:
[tex]\[
24 + 51.84592558726288
\][/tex]
Hence, the simplified form of the given expression is:
[tex]\[
(24, 51.84592558726288)
\][/tex]
