Question

Find the value of m and n: if: (5+2√3)/(7+4√3) = m + n√3

Answer 1

$\:\:\:\: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \huge\mathcal{\bf{\underline{\underline{\huge\mathcal{Answer}}}}}$

$\large\mathcal\red{solution}$

$\frac{5 + 2 \sqrt{3} }{7 - 4 \sqrt{3} } = m + n \sqrt{3} \\ = > \frac{ (5 + 2 \sqrt{3} )(7 + 4 \sqrt{3} )}{(7 - 4 \sqrt{3})(7 + 4 \sqrt{ 3} ) } = m + n \sqrt{3} \\ = > \frac{5(7 + 4 \sqrt{3}) + 2 \sqrt{3} (7 + 4 \sqrt{3} ) }{7 {}^{2} - (4 \sqrt{3} ) {}^{2} } = m + n \sqrt{3} \\ = > \frac{35 + 20 \sqrt{3} + 14 \sqrt{3} + 24}{49 - 48} =m + n \sqrt{3} \\ = > 59 + 34 \sqrt{3} = m + n \sqrt{3} \\ compairing \: both \: sides \: \: \\....... m = 59 \\ \\ ........n = 34$

$\large\mathcal\red{hope\: this \: helps \:you......}$

Answer 2

Step-by-step explanation:

(5+2√3)/(7+4√3)

= (5+2√3) (7-4√3)

--------------- × ------------

(7+4√3) 7-4√3

(5+2√3) × (7-4√3)

= ------------------------------

49 - 16*3

= 59 + 34√3

59 + 34√3 = m + n√3

By comparison, m= 59 & n = 34