Question

please give me answer of this question​

Answer 1

Answer:

a = 11, b = - 6

Step-by-step explanation:

Given : ${\sf{\ \ {\dfrac{ 5 + 2 {\sqrt{3}} }{ 7 + 4 {\sqrt{3}} }} = a + b {\sqrt{3}}}}$

To Find : ${\sf{\ \ a \ \& \ b}}$

Solution :

L.H.S. = ${\sf{\ \ {\dfrac{ 5 + 2 {\sqrt{3}} }{ 7 + 4 {\sqrt{3}} }}}}$

___________________________

• Rationalising the denominator.

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

___________________________

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

___________________________

${\boxed{\tt{Identity \ : \ (a - b)(a + b) = a^2 - b^2}}}$

${\tt{Here, \ a = 7, \ b = 4 {\sqrt{3}} }}$

___________________________

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

___________________________

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

___________________________

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

___________________________

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

___________________________

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

___________________________

$\implies{\sf{5(7 - 4 {\sqrt3}} ) + 2 {\sqrt{3}} (7 - 4 {\sqrt{3}} )}}$

___________________________

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

___________________________

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

___________________________

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

___________________________

• Writing common terms together.

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

___________________________

$\implies{\sf{ 11 - 6 {\sqrt{3}} }}$

___________________________

• We can write this as :

$\implies{\sf{(11) + (- 6) {\sqrt{3}} }}$

___________________________

• On comparing this with a + b√3, we get

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

___________________________