Question

Find
the value
of
a &b.
5+2√3÷7+4√3=a+b√3
​

Answer 1

$\frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \\ = \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } \\ = \frac{5(7 - 4 \sqrt{3} ) + 2 \sqrt{3}(7 - 4 \sqrt{3}) }{ {(7)}^{2} - {(4 \sqrt{3} )}^{2} } \\ = \frac{45 - 20 \sqrt{3} + 14 \sqrt{3} - 8 \times 3}{49 - 16 \times 3} \\ = \frac{45 - 24 - 6 \sqrt{3} }{49 - 48} \\ = \frac{21 - 6 \sqrt{3} }{1} \\ = 21 - 6 \sqrt{3}$

a=21

b= -6

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Answer 2

Answer:

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Step-by-step explanation:

Hi ,

LHS = (5 + 2√3 ) / ( 7 + 4√3 )

rationalize the denominator

= (5 + 2√3 ) ( 7 - 4√3 ) / [ ( 7 + 4√3 ) ( 7 - 4√3 ) 

= [ 5 ×7 - 5 × 4√3 + 2√3 × 7 - 2√3 × 4√3 ] / [ (7 )² - (4√3 )² ]

here we used ( x + y ) (x - y ) = x² - y²  identity

= [35 -20√3 + 14√3 -24 ] / [ 49 - 48 ]

= (11 - 6√3 )

therefore ,

LHS = RHS

11 - 6√3 = A - B√3

comparing both sides

A = 11,

B = 6

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