Question

if a and b are rational numbers and 3+2✓7/5+3✓7 find the value of a and b​

Answer 1

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Answer: a=11, b=-6

Solution:

Given (5 + 2√3)/(7 + 4√3) = a + b√3

Rationalizing the denominator on left-hand-side by multiplying the numerator and denominator with (7 - 4√3),

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

Multiply term by term the two expressions on numerator of L.H.S. and for the denominator apply the identity (m+n) (m-n) = m² - n² . We obtain,

(35 - 20√3 + 14√3 - 8.√3.√3)/[7² - (4√3)²] = a + b√3

Or, (35 - 6√3 - 8.3)/(49 - 48) = a + b√3

Or, (35 - 6√3 - 24)/1 = a + b√3

Or, 11 - 6√3 = a + b√3

Now equate the rational and irrational terms from both sides.

11 = a

Or, a = 11

- 6√3 = b√3

⇒ b = -6

Verification:

To prove (5 + 2√3)/(7 + 4√3) = a + b√3

i.e. to prove (5 + 2√3) = (a + b√3) (7 + 4√3)

Substituting for a=11 and b=-6,

R.H.S.= (a + b√3) (7 + 4√3)

= (11 - 6√3) (7 + 4√3) = 11.7 + 11.4√3 - 6√3.7 - 6.4.√3.√3 = 77 + 44√3 - 42√3 - 24.3

= 77 + 2√3 - 72 = 5 + 2√3 = L.H.S.