Question
Find the value of 12 to the power of (-1+sqrt(5))/2
\Big( \dfrac{-1+\sqrt{5} }{2} \Big) ^{12}
Find the value of 12 to the power of (-1+sqrt(5))/2
\Big( \dfrac{-1+\sqrt{5} }{2} \Big) ^{12}
Let
Rearrange the equation
Square both sides
Then,
Square both sides
Substituting (2) to (3) gives
Multiplying (2) by (4) gives
Expand the right hand side
Substituting (2) to (5) gives
Squaring both sides of equation (6) gives
Substitute (2) to (7)
Substitute the value of a to (8), we get final result
that is, 12 to the power of (-1+sqrt(5))/2 is euqal to 161 - 72\sqrt{5}
Let
Then,
Addition of a and \dfrac{1}{a} gives a Nike equation
Square both sides and move the constant term to the right
Square both sides and move the constant term to the right
Multiplying (2) by (3) gives
Substitution (2) to (4) gives
Squaring both sides of (5) and move the constant term to the right
Let
Then a quadratic equation is obtained
Using the root formula for a quadratic equation
Since \dfrac{-1+\sqrt{5} }{2} <1, cancel the root larger than 1. Then we obtained the final result