If the value of x+1/x=2 then what is the value of x^2014+1/x^2016
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To find the value of x^2014 + 1/x^2016 given that x + 1/x = 2, we can use the properties of exponents and algebraic manipulation. Let's break it down into steps:
Step 1: Simplify the expression x^2014 + 1/x^2016 by multiplying both terms by x^2016:
(x^2014 * x^2016) + (1/x^2016 * x^2016)
x^4024 + 1
Step 2: Substitute the value of x + 1/x in terms of x^2:
x + 1/x = 2
Multiply through by x:
x^2 + 1 = 2x
Rearrange the equation:
x^2 - 2x + 1 = 0
Step 3: Solve the quadratic equation x^2 - 2x + 1 = 0 by factoring or using the quadratic formula.
By factoring:
(x - 1)(x - 1) = 0
x - 1 = 0
x = 1
Step 4: Now substitute the value of x back into the simplified expression from step 1:
x^4024 + 1 = 1^4024 + 1 = 1 + 1 = 2
Therefore, the value of x^2014 + 1/x^2016, when x + 1/x = 2, is 2.