Question

Express 7-3 root 13/ 11-2 root 13 in the form a+b root 13

Answer 1

Question

Find the value of a and b where

$\rm \: \frac{7 - 3 \sqrt{13} }{11 - 2 \sqrt{13} } = a + b \sqrt{13}$

Solution:-

using rationalization methods

$\dfrac{7 - 3 \sqrt{13} }{11 - 2 \sqrt{13} }$

$\rm \: \dfrac{7 - 3 \sqrt{13} }{11 - 2 \sqrt{13} } \times \dfrac{11 + 2 \sqrt{13} }{11 + 2 \sqrt{13} }$

Using this identity

$\rm \: \to \: (a - b)(c + d) = (ac + ad - bc - bd)$

$\rm \: \to \: (a - b)(a + b) = ( {a}^{2} - {b}^{2} )$

Apply this identity

$\rm \: \dfrac{7 \times 11 + 7 \times 2 \sqrt{13 } - 11 \times 3 \sqrt{13} - 3 \sqrt{13} \times 2 \sqrt{13} }{(11) {}^{2} - (2 \sqrt{13} ) {}^{2} }$

$\rm \: \dfrac{77 + 14 \sqrt{13} - 33 \sqrt{13} - 6 \times 13 }{121 - 4 \times 13}$

$\rm \: \dfrac{77 - 19 \sqrt{13} - 78}{121 - 52}$

$\rm \dfrac{ - 1 - 19 \sqrt{13} }{69}$

So

$\rm \: a = \frac{ - 1}{69} \: \: and \: b = \frac{ - 19}{69}$