Question

if 5 +2√3 / 7 + 4√3 find the value of a^2 + b^2 ​

Answer 1

$11 \: and \: - 6 \: is \: the \: value \: of \: a \: and \: b \: if \\ \bold{\frac{(5+2 \sqrt{3})}{(7+4 \sqrt{3})}=a-b \sqrt{3}.}$

$Given:</p><p> \\ </p><p>\frac{(5+2 \sqrt{3})}{(7+4 \sqrt{3})}=a-b \sqrt{3}(7+43)(5+23)=a−b3</p><p></p><p>$

$To find: \\ </p><p></p><p>The \: value \: of \: a \: and \: b \: .= \: ?</p><p></p><p>Solution:</p><p> \\ \: </p><p>To \: find \: the \: value \: of \: a \: and \: b, \: rationalize \: the \: given \: fraction \: with  \: \\ 7- 4\sqrt{3}7−43 . \\ By \: multiplying \: it \: both \: in \: numerator \: and \: denominator \: we \: get: \: \\ </p><p></p><p>\frac{(5+2 \sqrt{3})}{(7+4 \sqrt{3})} \cdot \frac{(7-4 \sqrt{3})}{(7-4 \sqrt{3})}=\frac{5 \times 7-5 \times 4 \sqrt{3}+2 \sqrt{3} \times 7-2 \sqrt{3} \times 4 \sqrt{3}}{7^{2}-4^{2} \cdot 3}(7+43)(5+23)⋅(7−43)(7−43)=72−42⋅35×7−5×43+23×7−23×43</p><p></p><p>Solving \: the \: denominator \: in \: form \: of  \: (a^2-b^2)(a2−b2) \:  to \: get \: the \: value \: of \: the \: denominator \: </p><p></p><p>\frac{5 \times 7-5 \times 4 \sqrt{3}+2 \sqrt{3} \times 7-2 \sqrt{3} \times 4 \sqrt{3}}{7^{2}-4^{2} \cdot 3}72−42⋅35×7−5×43+23×7−23×43</p><p></p><p>Subtracting \: twenty \: four \: from \: thirty \: five \: and \: -20\sqrt{3} + 14\sqrt{3}203+143 separately \: we \: get \\ </p><p></p><p>\frac{[35-20 \sqrt{3}+14 \sqrt{3}-24]}{[49-48]}=11-6 \sqrt{3}[49−48][35−203+143−24]=11−63</p><p></p><p>Therefore, \: the \: value \: of \: the \: rationalization \: is \:  11-6 \sqrt{3}11−63</p><p></p><p>Now equating \bold{11-6 \sqrt{3} with a-b \sqrt{3}}11−63witha−b3,we get the value of \bold{a = 11}a=11 and \bold{b = - 6.}b=−6.</p><p></p><p>$