Question

If a and b are rational numbers and 7 minus 4 root 3 by 7 + 4 root 3 is equal a + b root 3 find the values of a and b

Answer 1

Identities used :

${(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy \\ (x + y)(x - y) = {x}^{2} - {y}^{2}$

$\frac{7 - 4 \sqrt{3} }{7 + 4 \sqrt{3} } = a + b \sqrt{3} \\ \\ \bf \: L.H.S \\ \\ \bf \: On \: rationalizing \: the \: denominator \\ \bf \: we \: get \\ \\ = \frac{7 - 4 \sqrt{3} }{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } \\ \\ = \frac{ {(7)}^{2} + {(4 \sqrt{3}) }^{2} - 2(7)(4 \sqrt{3} )}{ {(7)}^{2} - {(4 \sqrt{3} )}^{2} } \\ \\ = \frac{49 + 48 - 56 \sqrt{3} }{49 - 48} \\ \\ = 97 - 56 \sqrt{3} \\ \\ \bf \: On \: comparing \: L.H.S \: and \: R.H.S \\ \bf \: we \: get \\ \\ 97 - 56 \sqrt{3} = a + b \sqrt{3} \\ \\ \bf \: a = 97 \: \: : b = - 56$