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

$\mathfrak{\underline{\underline{Answer:}}}$

• m = 59
• n = 34

$\mathfrak{\underline{\underline{Explanation:}}}$

Given:

$\rm \dfrac{5 + 2 \sqrt{3} }{7 - 4 \sqrt{3} } = m + n \sqrt{3}$

To find: Values of m and n

Solution:

LHS : $\rm \dfrac{5 + 2 \sqrt{3} }{7 - 4 \sqrt{3} }$

Rationalise the denominator

$\implies \rm \dfrac{5 + 2 \sqrt{3} }{7 - 4 \sqrt{3} } \times \dfrac{7 + 4 \sqrt{3} }{7 + 4 \sqrt{3} } \\ \\ \implies\frac{(5 + 2 \sqrt{3} )(7 + 4 \sqrt{3} )}{ {7}^{2} - {(4 \sqrt{3} )}^{2} } \\ \\ \implies \frac{{5(7 + 4 \sqrt{3} )} + 2 \sqrt{3}(7 + 4 \sqrt{3} )}{49 - 48} \\ \\ \implies{ \frac{35 + 20 \sqrt{3} + 14 \sqrt{3} + 24 }{1} } \\ \\ \rm <strong>LHS</strong>= 59 + 34 \sqrt{3}$

$\rm{<strong>RHS</strong> = m + n \sqrt{3} }$

On comparing LHS and RHS we get,

m = 59

n = 34