Find the value of a if
√5+2√3 / 7+4√3
= a 6√3 .
Answers
Answer:
Lhs = (5+2√3)/(7+4√3)
= [(5+2√3)(7-4√3)]/[(7-4√3)(7+4√3)]
=[35-20√3+14√3-24]/[7²-(4√3)²]
=[11-6√3]/[49-48]
=11-6√3
therefore
11-6√3 = a+b√3
compare both sides
a= 11, b= -6
Step-by-step explanation:
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Answer:
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.
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Solving the LHS,
(5+2√3)/(7+4√3).
Here, since the denominator is not a rational number, we can rationalise it for our ease by multiplying (7–4√3) with the numerator as well as denominator such that we form (a+b)*(a-b) which would be same as a²-b² which is a popular identity.
After multiplying, we would get
(5+2√3)*(7–4√3)/(49–48)=(35–20√3+14√3–24)/1
=11–6√3 or 11+(-6)√3.
Comparing it to RHS , we get a=11 and b=-6.
Thank you.
First simplify (5+2√3)/(7+4√3)
=((5+2√3)(7–4√3))/(7^2-(4√3)^2)
=(35+14√3–20√3–24)/(49–48)
=11–6√3
Now,
(5+2√3)/(7+4√3)=a+b√3
Or,11–6√3=a+b√3
Therefore a=11 & b=-6.
What is the value of a and b, if (3+√5) / (3-√5) =a+b√5 where a and b are rational numbers?
What is the answer of simplify 7+3√5\3+√5+7-3√5\3-√5?
What will be value of √ (-√3+√ (3+8√ (7+4√3)))?
(5+2√3)/(7+4√3)=a+b√3
Rationalising the denominator gives
(5+2√3)(7–4√3)/{(7+4√3)(7–4√3)}=a+b√3
➝{5× (7-4√3)+2√3 ×(7-4√3)}/{ 49–48}=a+b√3
{5×7-5×4√3+7× 2√3-2 √3×4√3}=a+b √ 3
35-20√3+14√3-24=a+b√3
➝11- 6√3=a+b√3
Hence a=11
and b√3=-6√3
Hence b=-6
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Multiplying in Nr and Dr by (7–4√3)
(5+2√3)×(7–4√3)/(7+4√3)×(7–4√3)= a+b√3
or. (35–6√3–24)/(49–48) = a+b√3
or. 11–6√3 = a+b√3
Comparing the both sides.
a=11. and b= -6. Answer.
5 + 2√3/7 + 4√3 = a + b√3
=> (35 + 2√3 +28√3) /7 = a + b√3
=> 35 + 30√3 = 7a + 7b√3
By equating coefficients
7a = 35
=> a = 5
& 7b = 30
=> b = 30/7
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So,
a = 11 and b = -6.
I hope this ans. is satisfying. I left a few calculation purposely. I feel if you are doing bits of calculation in your brain then you are engaging your brain to do something plus I didnt want to spoonfeed every lil detail.