Let's analyze the given options step-by-step to determine which expression simplifies to [tex]\(4 \sqrt{13}\)[/tex].
First, we need to find the numerical value of [tex]\(4 \sqrt{13}\)[/tex]:
[tex]\[ 4 \sqrt{13} \][/tex]
We will compare this value to the numerical values of the given options:
1. [tex]\(\sqrt{29}\)[/tex]
2. [tex]\(\sqrt{17}\)[/tex]
3. [tex]\(\sqrt{52}\)[/tex]
4. [tex]\(\sqrt{208}\)[/tex]
Next, let's compute the approximate value of each of these square roots:
1. [tex]\(\sqrt{29}\)[/tex] is approximately [tex]\(5.385\)[/tex]
2. [tex]\(\sqrt{17}\)[/tex] is approximately [tex]\(4.123\)[/tex]
3. [tex]\(\sqrt{52}\)[/tex] is approximately [tex]\(7.211\)[/tex]
4. [tex]\(\sqrt{208}\)[/tex] is approximately [tex]\(14.422\)[/tex]
Now, let's identify which of these computed values matches the value of [tex]\(4 \sqrt{13}\)[/tex]:
- [tex]\(4 \sqrt{13}\)[/tex] corresponds approximately to [tex]\(14.422\)[/tex]
Upon reviewing our calculations, we find that:
[tex]\[ \sqrt{208} \approx 14.422 \][/tex]
Therefore, the correct option that simplifies to [tex]\(4 \sqrt{13}\)[/tex] is:
[tex]\[ \sqrt{208} \][/tex]
Hence, the answer is:
[tex]\[ \boxed{\sqrt{208}} \][/tex]
