The Arithmetic mean of two surds is
5 + 4√3
and one of them is
2 + 4√3
then the
square root of the second surd is
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The Arithmetic mean of two surds is
5 + 4√3
and one of them is
2 + 4√3
then the
square root of the second surd is
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Step-by-step explanation:
Given :-
The Arithmetic mean of two surds is
5 + 4√3 and one of them is 2 + 4√3
To find :-
Find the square root of the second ssurd?
Solution :-
Given that :
The Arithmetic mean of two surds = 5 + 4√3
One of the surds = 2 + 4√3
Let the other surd be X
Arithmetic Mean of the two numbers a and b is (a+b)/2
=> The Arithmetic mean of (2+4√3) and X
=> (2+4√3+X)/2
According to the given problem
The Arithmetic mean of two surds = 5 + 4√3
=> (2+4√3+X)/2 = 5 + 4√3
=> (2+4√3+X) = 2(5+4√3)
=> 2+4√3+X = 10+8√3
=> X = (10+8√3)-(2+4√3)
=>X = 10+8√3-2-4√3
=> X = (10-2) +(8√3-4√3)
=>X = 8+4√3
The other surd = 8+4√3
The square root of 8+4√3
=>√(8+4√3)
=> √(8+2(√3×2×2))
=>√(8+2√12)
=> √[(6+2)+2(√6×√2)]
=>√[(√6)^2+(√2)^2+2(√6×√2)]
It is in the form of a^2+2ab+b^2
Where a = √6 and b=√2
We know that
a^2+2ab+b^2 = (a+b)^2
=> √[(√6)^2+(√2)^2+2(√6×√2)]
=>√[(√6+√2)^2]
=>√6+√2
Therefore, √(8+4√3) = √6+√2
Answer:-
The square root of 8+4√3= √6+√2
Used formulae:-