Using the root formula for a quadratic equation,

the roots for the equation

x^2-x-1 = 0
(1)

can be determined as

x=\dfrac{1\pm\sqrt{5} }{2}
(2)

which are the Golden Ratios. The equation (1) is often referred as Golden Equation.

To obtain the value of x to the power of 6, we can do the following manipulation on Golden Equation.

First, moving -x-1 to the right hand side yields the expression represent x to the power of 2

x^2 = x+1
(3)

Addition 2 to both sides of the Golden Equation

x^2-x+1 = 2

And then, multiply x+1 on both sides.

(x+1)(x^2-x+1) = 2(x+1)

Now we can use difference of cubes to raise x to the power of 3

x^3+1 = 2x+2

Then

x^3 = 2x+1

Squaring both sides gives the expression to represent x to the power of 6

\begin{aligned} x^6 &= (2x+1)^2 \\ &=4x^2+4x+1 \\ \end{aligned}
(4)

Now substitute x^2 in (3), and the value of x in (2) to (4)

\begin{aligned} x^6 &= (2x+1)^2 \\ &=4(x+1)+4x+1 \\ &=8x+5\\ &=8\bigg( \dfrac{1\pm\sqrt{5} }{2} \bigg) +5 \\ &=9\pm4\sqrt{5} \end{aligned}
(5)

So the final result of the value of x^6 is 9\pm4\sqrt{5}

Steven Zheng posted 1 hour ago

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