﻿ Find the value of 12 to the power of (1+sqrt(5))/2 \Big( \dfrac{\sqrt{5}+1 }{2}\Big) ^{12}

#### Question

Find the value of 12 to the power of (1+sqrt(5))/2

\Big( \dfrac{\sqrt{5}+1 }{2}\Big) ^{12}

Collected in the board: Nike function

Steven Zheng posted 4 months ago

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}

Steven Zheng posted 4 months ago

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}

Steven Zheng posted 4 months ago

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