Explanation:
Given that: √[-√(3)+√{3+8√(7+4√(3))}]
= √[-√(3) + √{3+8√((√3)² + √(4)² + 2√(4*3))}]
= √[-√(3) + √{3 + 8√(√(3) + √(4))²}] [Since, {√(a) + √(b)}² = a+b + 2√(ab)]
= √[-√(3) + √{3 + 8√(√(3) + √(4))²}]
= √[-√(3) + √{3 + 8(√(3) + √(4))}]
= √[-√(3) + √{3 + 8(√(3) + √(2*2))}]
= √[-√(3) + √{3 + 8(√(3) + 2)}]
= √[-√(3) + √{3 + 16 + 8√(3)}]
= √[-√(3) + √{19 + 8√(3)}]
= √[-√(3) + √{16 + 3 + 2√(3*16)}]
= √[-√(3) + √{(16)² + (3)² + 2√(3*16)}]
= √[-√(3) + √{√(16)² + √(3)²}] [Since, {√(a) + √(b)}² = a+b + 2√(ab)]
= √{-√(3) + √(16) + √(3)}
= √{-√(3) + √(4*4) + √(3)}
= √{-√(3) + 4 + √(3)}
Now, cancel -√3 and +√3.
= √(4)
= √(2*2)
= 2
Answer: Hence, the simplified form of √[-√(3)+√{3+8√(7+4√(3))}] = 2.
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