Question

Find the value of a in the following :
5+2√3 / 7+4√3 = a - 6√3​

Answer 1

Answer:

a = 11 and b = - 6

Explanation:

Answer 2

$\underline{ \large{ \sf Solution - }}$

$\sf \implies \: \dfrac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = a - 6 \sqrt{3}$

Here, we need to find : Value of a = ?

On taking LHS,

$\sf \implies \: \dfrac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} }$

Multiplying the conjugate of the denominator to the fraction.

$\sf \implies \: \dfrac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \times \dfrac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} }$

Combine the fractions,

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

We know that,

$\sf \implies \: \boxed{\sf (a + b) (a - b) = (a)^{2} - (b) ^{2} }$

So,

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

$\sf \implies \: \dfrac{(5 + 2 \sqrt{3}) (7 - 4 \sqrt{3})}{49 - 48 }$

$\sf \implies \: \dfrac{(5 + 2 \sqrt{3}) (7 - 4 \sqrt{3})}{1}$

$\sf \implies \: (5 + 2 \sqrt{3}) (7 - 4 \sqrt{3})$

$\sf \implies \: 5 \times 7 + 5(- 4 \sqrt{3}) + 2 \sqrt{3} \times 7 + 2 \sqrt{3} ( - 4 \sqrt{3} )$

$\sf \implies \: 35 - 20 \sqrt{3} + 14 \sqrt{3} - 24$

$\sf \implies \: 11- 20 \sqrt{3} + 14 \sqrt{3}$

$\bf \implies \: 11- 6 \sqrt{3}$

Then,

$\sf \implies \: 11- 6 \sqrt{3} = a - 6 \sqrt{3}$

Thus,

• Value of a = 11 ★