Question

If a and b are rational numbers and 5+3√3 7+4√3 = a + b√3, then find the values of a and b.

Answer 1

Answer:

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

Step-by-step explanation:

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