Given that: √{11 + 4√(7)}
Let √{11 + 4√(7)} = √(x) + √(y)
Squaring on both sides then
= [√{11 + 4√(7)}]² = {√(x) + √(y)}²
= 11 + 4√(7) = x + y + 2√(xy)
= 11 + 2√(7 * 2²) = x + y + 2√(xy)
= 11 + 2√(7 * 2*2) = x + y + 2√(xy)
= 11 + 2√(7 * 4) = x + y + 2√(xy)
= 11 + 2√(28) = x + y + 2√(xy)
On comparing with both sides, we notice that:
⇛2√(xy) = 2√(28)
⇛ xy = 28
We know that
(x-y)² = (x+y)² - 4xy
(x-y)² = (11)² - 4(28)
(x-y)² = (11*11) - 112
(x-y)² = 121 - 112
(x-y)² = 9
x - y = √(9)
x - y = √(3*3)
On adding equation (1) and equation (2), we get
x + y = 11
(+) x - y = 3
_______
2x = 14
Shift the number 2 from LHS to RHS, changing it's sign.
⇛x = 14/2
⇛x = 7/1
Simplify the fraction to get the value of x.
Now, substitute the value of x in equation (1) then
x + y = 11
⇛7 + y = 11
Shift the number 7 from LHS to RHS, changing it's sign.
⇛y = 11 - 7
Now, we have: x = 7 and y = 4
So,
√(x) + √(y) = √(7) + √(4) = √(7) + √(2*2) √(7) + 2
(Or)
√{11 + 4√(7)}
= √{11 + 2√(7*4)}
= √{11 + 2√(7)√(4)}
= √[{√(7)²} + {√(4)}² + 2√(7)√(4)
[since, (a+b)² = a² + 2ab + b²]
= √[{√(7) + √(4)}²]
= √(7) + √(4)
= √(7) + √(2*2)
Therefore, √{11 + 4√(7)} = √(7) + 2
Answer: Hence, the simplified form of √{11 + 4√(7)} is √(7) + 2
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The solution √(11 +4 √7) is mathematically given as
x=4.64575131
Question Parameters:
√(11 +4 √7)
The question requires resolving the above expression and deciphering the solution
Generally, the equation is mathematically given as
x=√(11 +4 √7)
Hence
x=√(11 +4*2.64575131)
In conclusion, The resolution of the expression x=√(11 +4 √7) is
x=4.64575131
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