Certainly! Let's break down the problem step-by-step to find [tex]\(\sqrt{5 \sqrt{5 \sqrt{5 \sqrt{5 \sqrt{5}}}}}\)[/tex].
1. Start with the innermost nested square root:
[tex]\[
\sqrt{5}
\][/tex]
According to the previous results, this value is approximately 2.236.
2. Substitute this result into the next nested square root:
[tex]\[
\sqrt{5 \sqrt{5}} = \sqrt{5 \times 2.236} = \sqrt{11.180}
\][/tex]
The value of [tex]\(\sqrt{11.180}\)[/tex] simplifies to approximately 1.495.
3. Substitute the new result into the next nested square root:
[tex]\[
\sqrt{5 \sqrt{5 \sqrt{5}}} = \sqrt{5 \times 1.495} = \sqrt{7.475}
\][/tex]
The value of [tex]\(\sqrt{7.475}\)[/tex] simplifies to approximately 1.223.
4. Substitute this result into yet another nested square root:
[tex]\[
\sqrt{5 \sqrt{5 \sqrt{5 \sqrt{5}}}} = \sqrt{5 \times 1.223} = \sqrt{6.115}
\][/tex]
The value of [tex]\(\sqrt{6.115}\)[/tex] simplifies to approximately 1.106.
Therefore, the value of [tex]\(\sqrt{5 \sqrt{5 \sqrt{5 \sqrt{5 \sqrt{5}}}}}\)[/tex] is approximately 1.106.
