To simplify the expression [tex]\(\sqrt[4]{\sqrt[3]{2^2}}\)[/tex], we should work through it step-by-step, starting from the inside and working our way out. Let's break it down:
1. Evaluate the innermost expression [tex]\(2^2\)[/tex]
[tex]\[
2^2 = 4
\][/tex]
So, the expression now becomes:
[tex]\[
\sqrt[4]{\sqrt[3]{4}}
\][/tex]
2. Calculate the cube root of 4
[tex]\[
\sqrt[3]{4} \approx 1.5874010519681994
\][/tex]
Therefore, the expression now is:
[tex]\[
\sqrt[4]{1.5874010519681994}
\][/tex]
3. Finally, calculate the fourth root of [tex]\(1.5874010519681994\)[/tex]
[tex]\[
\sqrt[4]{1.5874010519681994} \approx 1.122462048309373
\][/tex]
So, after simplifying the given expression [tex]\(\sqrt[4]{\sqrt[3]{2^2}}\)[/tex], the final result is approximately [tex]\(1.122462048309373\)[/tex].
