Question

51. If a =√(7+4√3/7-4√3) then the value of [a(a-14)^2 is
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Answer 1

Answer:

-1

Simplify a

$a = \sqrt{ \frac{7 + 4 \sqrt{3} }{7 - 4 \sqrt{3} } \times \frac{7 + 4 \sqrt{3} }{7 + 4 \sqrt{3} } } \\ \\ = \sqrt{ \frac{ ({7 + 4 \sqrt{3}) }^{2} }{ {7}^{2} - (4 \sqrt{3}) {}^{2} } } \\ \\ = \frac{7 + 4 \sqrt{3} }{ \sqrt{49 - 48} } \\ \\ = 7 + 4 \sqrt{3}$

So

$a(a - 14) {}^{2} \\ = (7 + 4 \sqrt{3} )(7 + 4 \sqrt{3} - 14) \\ = (4 \sqrt{ 3} + 7)(4 \sqrt{3} - 7) \\ = ( {4 \sqrt{3} )}^{2} - {7}^{2} \\ = 48 - 49 \\ = - 1$