Question

Find the value of 'a' and 'b' if :

Answer 1

Answer:-

Given (5 + 2√3)/(7 + 4√3) = a + b√3

Rationalizing the denominator on left-hand-side by multiplying the numerator and denominator with (7 - 4√3),

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

Multiply term by term the two expressions on numerator of L.H.S. and for the denominator apply the identity (m+n) (m-n) = m² - n² . We obtain,

(35 - 20√3 + 14√3 - 8.√3.√3)/[7² - (4√3)²] = a + b√3

Or, (35 - 6√3 - 8.3)/(49 - 48) = a + b√3

Or, (35 - 6√3 - 24)/1 = a + b√3

Or, 11 - 6√3 = a + b√3

Now equate the rational and irrational terms from both sides.

11 = a

Or, a = 11

$- 6√3 = b√3$

$⇒ b = -6$

Answer 2

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$\frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \\ = \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } \\ = \frac{5 + 2 \sqrt{3} \times 7 - 4 \sqrt{3} }{(7) {}^{2} - (4 \sqrt{3)} {}^{2} } \\ = \frac{35 + 14 \sqrt{3} - 20 \sqrt{3} - 8 \sqrt{3} \sqrt{3} }{49 - 16 \sqrt{3} \sqrt{3} } \\ = 35 - \sqrt{3} (14 - 8) - 24 \\ = 35 - 3 \sqrt{6} - 24 \\ = 11 - 6 \sqrt{3} \\ \\ on \: comparing \: we \: get \: \\ = a \: = 11 \\ = b = - 6$

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