Step-by-step explanation:

•Appropriate Question • :-

If (√7+√3)/(√7-√3) = a+b√21 then find a and b

• Solution • :-

Given that :

(√7+√3)/(√7-√3) = a+b√21

On taking (√7+√3)/(√7-√3)

The denominator = √7-√3

The rationalising factor of √7-√3 is √7+√3

On rationalising the denominator then

=>[ (√7+√3)/(√7-√3) ]×[(√7+√3)/(√7+√3) ]

=> [(√7+√3)(√7+√3) ]/[(√7-√3)(√7+√3) ]

=> (√7+√3)²/[(√7)²-(√3)²]

Since, (a+b)(a-b) = a²-b²

Where, a = √7 and b = √3

=> (√7+√3)²)/(7-3)

=> (√7+√3)²/4

=> [(√7)²+2(√7)(√3)+(√3)²]/4

=> (7+2√21+3)/4

=> (10+2√21)/4

=> 2(5+√21)/4

=> (5+√21)/2

Now,

(√7+√3)/(√7-√3) = a+b√21

=> (5+√21)/2 = a+b√21

=> (5/2)+(√21/2) = a+b√21

=> (5/2)+(1/2)√21 = a+b√21

On comparing both sides then

a = 5/2

b = 1/2

• Answer •:-

• The value of a = 5/2
• The value of b = 1/2

• Points To Know • :-

• The product of two irrational numbers is a rational number then the two irrational numbers are called rationalising factors to each other.
• The rationalising factor of √a-√b is √a+√b
• The rationalising factor of √a+√b is √a-√b
• (a+b)(a-b) = a²-b²
• (a+b)² = a²+2ab+b²