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Sagot :
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]
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]
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