Question

please answer both questions with explanations ​

Answer 1

Step-by-step explanation:

Solutions:-

i) Given that :

(3+√2)/(3-√2) = a+b√2

The denominator = 3-√2

The Rationalising factor of 3-√2 is 3+√2

On Rationalising the denominator then

=> [(3+√2)/(3-√2)]×[(3+√2)/(3+√2)] = a+b√2

=> [(3+√2)(3+√2)]/[(3-√2)(3+√2)] = a+b√2

=> (3+√2)²/[3²-(√2)²] = a+b√2

Since (a+b)(a-b) = a²-b²

Where, a = 3 and b =√2

=> (3+√2)²/(9-2) = a+b√2

=> (3+√2)²/7 = a+b√2

=> [(3²+2(3)(√2)+(√2)²]/7 = a+b√2

=> (9+6√2+2)/7 = a+b√2

=> (11+6√2)/7 = a+b√2

=> (11/7)+(6/7)√2 = a+b√2

On comparing both sides then

a = 11/7 and b = 6/7

ii)Given that :

(5+2√3)/(7+4√3) = a+b√3

The denominator = 7+4√3

The Rationalising factor of 7+4√3 is 7-4√3

On Rationalising the denominator then

=> [(5+2√3)/(7+4√3)]×[(7-4√3)/(7-4√3)] = a+b√3

=> [(5+2√3)(7-4√3)]/[(7+4√3)(7-4√3)] = a+b√3

=> [(5+2√3)(7-4√3)]/[7²-(4√3)²] = a+b√3

Since (a+b)(a-b) = a²-b²

Where ,a = 7 and b = 4√3

=> [(5+2√3)(7-4√3)]/[49-48] = a+b√3

=> [(5+2√3)(7-4√3)]/1= a+b√3

=> [(5+2√3)(7-4√3)]= a+b√3

=>[5(7-4√3)+2√3(7-4√3)] = a+b√3

=> (35-20√3+14√3-24) = a+b√3

=> 11-6√3= a+b√3

On comparing both sides then

a = 11 and b = -6

Answer:-

i) a = 11/7 and b = 6/7

ii) a = 11 and b = -6

Used formulae:-

• The Rationalising factor of a-√b is a+√b
• The Rationalising factor of a+√b is a-√b
• (a+b)(a-b) = a²-b²