Sure, let's find the value of [tex]\(\sqrt[4]{\sqrt{72}}\)[/tex], which we also know can be written as [tex]\(\sqrt[16]{72}\)[/tex].
First, let's break down the problem step by step:
1. We start with [tex]\(\sqrt[4]{\sqrt{72}}\)[/tex]. To simplify this, first find the square root of 72.
[tex]\[
\sqrt{72} = 8.48528137423857
\][/tex]
2. Next, we need to take the fourth root of this result:
[tex]\[
\sqrt[4]{8.48528137423857} = 1.7067368368450775
\][/tex]
Alternatively, we can solve [tex]\(\sqrt[4]{\sqrt{72}}\)[/tex] by directly expressing it as a single root. The expression can be rewritten by combining the roots into one:
[tex]\[
\sqrt[4]{\sqrt{72}} = \sqrt[16]{72}
\][/tex]
This is equivalent to raising 72 to the power of 1/16:
[tex]\[
72^{1/16} = 1.7067368368450775
\][/tex]
Both methods yield the same result.
So, the value of [tex]\(\sqrt[4]{\sqrt{72}}\)[/tex] or [tex]\(\sqrt[16]{72}\)[/tex] is:
[tex]\[
\boxed{1.7067368368450775}
\][/tex]
