Question

find a & b if 5+2√3 / 7+4√3 = a - b √3​

Answer 1

Required answer:

The values of "a" and "b" respectively are 11 and 6.

$\qquad$_____________

Step-by-step explanation:

$\pink{ \tt \dashrightarrow \: \: \: \dfrac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = a - b \sqrt{3}}$

$\:$

Firstly, rationalizing the denominator of the LHS :

$\green{\tt \dashrightarrow \: \: \: \dfrac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \times \bigg \{ \dfrac{7 -4 \sqrt{3} }{7 - 4 \sqrt{3} } \bigg \}}$

$\:$

Using (a – b) (a + b) = a² – b² for the denominator :

$\pink{ \tt \dashrightarrow \: \: \: \dfrac{5 \: (7 - 4 \sqrt{3} ) + 2 \sqrt{3} \: (7 - 4 \sqrt{3} )}{ {(7)}^{2} - {(4 \sqrt{3} )}^{2} } }$

$\:$

$\green{\tt \dashrightarrow \: \: \: \dfrac{35 - 20 \sqrt{3} + 14 \sqrt{3} - 24 }{49 - 48}}$

$\:$

$\pink{\tt \dashrightarrow \: \: \: \dfrac{ 11- 6 \sqrt{3} }{1} }$

$\:$

$\green{ \tt \dashrightarrow \: \: \: 11 - 6 \sqrt{3} }$

$\qquad$_____________

On comparing this to RHS, we get the values as :

$\pink{ \tt \dashrightarrow\: \: \: a = 11}$

$\:$

$\green{ \tt \dashrightarrow \: \: \: b = 6}$

Answer 2

Question :-

find a & b if 5+2√3 / 7+4√3 = a - b √3.

Solution :-

Given :

$\sf{ \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = a + b \sqrt{3} }$

Since, Rationalization of :

$\sf{ \frac{a + \sqrt{b} }{c + \sqrt{d} } = \frac{a + \sqrt{b} }{c + \sqrt{d} } \times \frac{c - \sqrt{d} }{c - \sqrt{d} } }$

Taking LHS to rationalization,

$\sf:\longmapsto{ \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4\sqrt{3} } }$

Using Identity : (a+b) (a-b) = a² - b²

$\sf:\longmapsto{ \frac{(5 + 2 \sqrt{3} )(7 - 4 \sqrt{3} )}{(7 + 4 \sqrt{3})(7 - 4 \sqrt{3} ) } } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{ \frac{5(7 - 4 \sqrt{3} ) + 2 \sqrt{3} (7 - 4 \sqrt{3}) }{(7) {}^{2} - (4 \sqrt{3} ) {}^{2} } } \\ \\ \sf:\longmapsto{ \frac{35 - 20 \sqrt{3} + 14 \sqrt{3} - 8(3) }{49 - 16(3)} } \: \: \: \: \\ \\ \sf:\longmapsto{ \frac{35 - 6 \sqrt{3} - 24 }{49 - 48} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{11 - 6 \sqrt{3} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Now, Compare both side

$\sf:\longmapsto{11 - 6 \: \cancel{\sqrt{3} }= a - b \: \cancel{\sqrt{3} } } \\ \\ \sf:\longmapsto{11 - 6 = a - b} \: \: \: \: \: \: \: \: \: \: \: \:$

After comparison we get,

• a = 11
• b = 6

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

Learn more from brainly :

Find the value of a & b if 5 + 2√3= a + b√3 ------------ 7 + 4√3..

https://brainly.in/question/20711320