Find the value of a and b 5+2√3/7+4√3= a+b.
Answers
Step-by-step explanation:
What is the value of "a" and "b" if (5+2√3)/(7+4√3)=a+b√3?
Sachidanand Das
Answered 2 years ago
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.