Question

5+2 root3 upon 7+4 root 3​

Answer 1

$\red{\bf {Answer:}}$

$\underline{\boxed{\sf\purple{a=11, b=-6}}}$

$\red{\bf {Solution:}}$

$\frac{ (5 + 2√3)}{ (7 + 4√3)}$$\underline{\boxed{\sf\purple{= a + b√3}}}$

Rationalizing the denominator on left-hand-side by multiplying the numerator and denominator with (7 - 4√3),

$\frac{(5 + 2√3) (7 - 4√3)}{(7 + 4√3) (7 - 4√3)}$ $\underline{\boxed{\sf\purple{= a + b√3}}}$

Multiply term by term the two expressions on numerator of L.H.S. and for the denominator apply the identity (m+n) (m-n) = m² - n² .

$\bold{We \:obtain,}$

$\frac{(35 - 20√3 + 14√3 - 8.√3.√3)}{[7² - (4√3)²]}$ $\underline{\boxed{\sf\purple{= a + b√3}}}$

$\bold{Or, }$

$\frac{(35 - 6√3 - 8.3)}{(49 - 48)}$$\underline{\boxed{\sf\purple{= a + b√3}}}$

$\bold{Or, }$

$\frac{(35 - 6√3 - 24)}{1}$$\underline{\boxed{\sf\purple{= a + b√3}}}$

$\bold{Or, }$

$\underline{\boxed{\sf\purple{11 - 6√3 = a + b√3}}}$

Now equate the rational and irrational terms from both sides.

$\boxed{\sf{11 = a}}$

$\bold{Or, }$

$\boxed{\sf{a = 11}}$

$\bold{- 6√3 = b√3}$

$\underline{\boxed{\sf\purple{⇒ b = -6}}}$

Thankyou :)

Refer the attachment for better understanding :)