Question

if P=
$\sqrt{16 + 8 \sqrt{3} } - \sqrt{21 - 12 \sqrt{3} }$
Find value of P





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Answer 1

$i ) \sqrt{16 + 8 \sqrt{3}} \\= \sqrt{ 16 + 2\sqrt{48}} \\= \sqrt{ (12+4) + 2\sqrt{12\times 4}} \\= \sqrt{ \sqrt{12}^{2} + \sqrt{4}^{2} + 2\times \sqrt{12} \times \sqrt{4} } \\= \sqrt{ ( \sqrt{12} + \sqrt{4} )^{2}}\\= \sqrt{12} + \sqrt{4} \\= \sqrt{12} + 2 \: --(1)$

$ii ) \sqrt{21 - 12 \sqrt{3}} \\= \sqrt{ 21 - 2\sqrt{108}} \\= \sqrt{ (12+9) - 2\sqrt{12\times 9}} \\= \sqrt{ \sqrt{12}^{2} + \sqrt{9}^{2} - 2\times \sqrt{12} \times \sqrt{9} } \\= \sqrt{ ( \sqrt{12} - \sqrt{9} )^{2}}\\= \sqrt{12} - \sqrt{9} \\= \sqrt{12} - 3 \: --(2)$

$Now , \red { P = \sqrt{16 + 8 \sqrt{3} } - \sqrt{21 - 12 \sqrt{3} }}$

$\implies P = \sqrt{12} + 2 - ( \sqrt{12} - 3 ) \\\blue { [ From \: (1) \:and \: (2) ] }$

$\implies P = \sqrt{12} + 2 - \sqrt{12} + 3 \\= 2 + 3 \\= 5$

Therefore.,

$\red { Value \: P } \green { = 5 }$

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