Question
Find the value of 12 to the power of (1+sqrt(5))/2
\Big( \dfrac{\sqrt{5}+1 }{2}\Big) ^{12}
Answer 1
Let
x = \dfrac{\sqrt{5}+1 }{2}
(1)
Square both sides
x^2 = \Big( \dfrac{\sqrt{5}+1 }{2}\Big) ^2
Expanding RHS
x^2 =\dfrac{6+2\sqrt{5} }{4} =\dfrac{3+\sqrt{5} }{2} =1+\dfrac{1+\sqrt{5} }{2}= 1+x
Then we get the important equation
x^2 = 1+x
(2)
Squaring the both sides gives
x^4 = (1+x)^2
Expand RHS,
x^4 = 1+2x+x^2
(3)
Substitute (2) to (4)
x^4 = 2+3x
(4)
Multiply (2) by (4)
x^6 = (1+x)(2+3x)
Expand RHS,
x^6 = 2+5x+3x^2
(5)
Substitute (2) to (5)
x^6 = 2+5x + 3(1+x)
Then,
x^6 = 5 + 8x
(6)
Square both sides of equation (6)
x^{12} = 25+80x+64x^2
(7)
Substitute (2) to (7)
x^{12} = 25+80x+64(1+x)
Then,
x^{12} = 89+144x
(8)
Substitute the value of x to (8)
x^{12} = 89+144(\dfrac{\sqrt{5}+1 }{2}) = \dfrac{89\cdot 2+144+144\sqrt{5} }{2}
=89+72+72\sqrt{5} = 161+72\sqrt{5}
Finally, we could conclude the result
\Big( \dfrac{\sqrt{5}+1 }{2}\Big) ^{12} = 161+72\sqrt{5}
Answer 2
Let
x = \dfrac{1+\sqrt{5} }{2}
(1)
Taking reciprocal of x gives
\dfrac{1}{x} =\dfrac{1}{ \dfrac{1+\sqrt{5} }{2}} = \dfrac{-1+\sqrt{5} }{2}
(2)
Addition of x and \dfrac{1}{x} gives Nike equation in terms x ,
x+\dfrac{1}{x} = \dfrac{1+\sqrt{5} }{2} + \dfrac{-1+\sqrt{5} }{2}
Then
x+\dfrac{1}{x} = \sqrt{5}
(3)
Square both sides of (3)
x^2+\dfrac{1}{x^2} +2 = 5
Then,
x^2+\dfrac{1}{x^2} = 3
(4)
Square both sides of (4) gives
x^4+\dfrac{1}{x^4} = 7
(5)
Square both sides of (5) gives
x^8+\dfrac{1}{x^8} = 47
(6)
Multiplying (5) by (6) gives
x^{12}+\dfrac{1}{x^4}+x^4+\dfrac{1}{x^{12}} = 46\times7
(7)
Substituting the result of (5) to (7) results in
x^{12}+ \dfrac{1}{x^{12}} = 46\times7
(8)
Let
m = x^{12}
(9)
Substitute m to (8) and rearrange the equation. We get a quadratic equation
m^2-46\times 7m+1=0
(10)
Using the root formula for a quadratic equation, we get
m = \dfrac{46\times7+\sqrt{(46\times7)^2-4} }{2} (\text{cancel negative root})
Simplifying the root gives the final result
m = 161+72\sqrt{5}
Now we have obtained the final result
\Big( \dfrac{1+\sqrt{5} }{2}\Big) ^{12} = 161+72\sqrt{5}