Question
If a = \dfrac{\sqrt{101} +1 }{2}
find the value of (a^3-26a-23)^7
Answer
a = \dfrac{\sqrt{101} +1 }{2}
Then
2a = \sqrt{101}+1
Rearrange the terms to make the term with square root on one side
2a-1 = \sqrt{101}
Square both sides
4a^2-4a+1 = 101
Then
4a^2-4a = 100
a^2 = a+25
(1)
Now let's reduce the degree of polynomial using the result
(a^3-26a-23)^7
=[a(a+25)-26a-23]^7
=(a^2+25a-26a-23)^7
=(a^2-a-23)^7
=(a+25-a-23)^7
=2^7
=128