To solve the expression [tex]\( \sqrt{41 - \sqrt{21 + \sqrt{19 - \sqrt{9}}}} \)[/tex], we will decompose it step-by-step:
1. Evaluate the innermost square root:
[tex]\[
\sqrt{9} = 3
\][/tex]
2. Proceed to the next level:
[tex]\[
19 - \sqrt{9} = 19 - 3 = 16
\][/tex]
3. Take the square root of the previous result:
[tex]\[
\sqrt{16} = 4
\][/tex]
4. Move up one more level:
[tex]\[
21 + \sqrt{16} = 21 + 4 = 25
\][/tex]
5. Take the square root of this result:
[tex]\[
\sqrt{25} = 5
\][/tex]
6. Now, consider the outermost expression:
[tex]\[
41 - \sqrt{21 + \sqrt{19 - \sqrt{9}}} = 41 - \sqrt{25} = 41 - 5 = 36
\][/tex]
7. Finally, take the square root of the last result:
[tex]\[
\sqrt{36} = 6
\][/tex]
Thus, the value of [tex]\( \sqrt{41 - \sqrt{21 + \sqrt{19 - \sqrt{9}}}} \)[/tex] is:
[tex]\[
\boxed{6}
\][/tex]
