Question

if both a and b are rational numbers find the values of a and b in the given equalities 5+3√3 upon 7+4√3= a+b√3​

Answer 1

$\mathrm{Question :\;\dfrac{5 + 3\sqrt{3}}{7 + 4\sqrt{3}}}$

$\mathrm{Multiplying\;and\;Dividing\;the\;above\;fraction\;with\;7 - 4\sqrt{3}}$

$\implies \dfrac{7 - 4\sqrt{3}}{7 - 4\sqrt{3}} \times \dfrac{5 + 3\sqrt{3}}{7 + 4\sqrt{3}}$

$\implies \dfrac{\big(5 + 3\sqrt{3}\big)\big(7 - 4\sqrt{3}\big)}{\big(7 + 4\sqrt{3}\big)\big(7 - 4\sqrt{3}\big)}$

●  Use Identity : (A + B)(A - B) = A² - B²

$\implies \dfrac{(5)(7) - 5(4\sqrt{3}) + (7)(3\sqrt{3}) - (4\sqrt{3})(3\sqrt{3})}{\big(7\big)^2 - \big(4\sqrt{3}\big)^2}$

$\mathrm{\implies \dfrac{35 - 20\sqrt{3} + 21\sqrt{3} - (4)(3)(\sqrt{3})^2}{49 - (4)^2(\sqrt{3})^2}}$

$\mathrm{\implies \dfrac{35 + \sqrt{3} - (12)(3)}{49 - (16)(3)}}$

$\mathrm{\implies \dfrac{35 + \sqrt{3} - 36}{49 - 48}}$

$\mathrm{\implies {-1 + \sqrt{3}}}$

$\mathrm{\implies -1 + \sqrt{3} = a + b\sqrt{3}}$

Comparing both sides :

●  a = - 1

●  b = 1

Answer 2

Answer: A = -1 and B =1

Step-by-step explanation:

$\frac{5+3\sqrt{3} }{7+4\sqrt{3} }$  

Rationalising the denominator by multiplying both denominator nd numerator with 7-4√3

$\frac{5+3\sqrt{3} }{7+4\sqrt{3} } \times \frac{7-4\sqrt{3} }{7-4\sqrt{3} }$

Using identity : (a+b) (a-b) = a^2 - ^2

$\frac{(5+3\sqrt{3} )(7-4\sqrt{3} )}{7^{2} - (4\sqrt{3}) ^{2} }$

$\frac{35-20\sqrt{3}+21\sqrt{3} -36 }{49 - 48}$

$\sqrt{3} -1 = a+ b\sqrt{3}$

So, a= -1 and B = 1