(a+1/a)^2=3 then a^6-1/a^6
Solution to the Equation
Given the equation (a+1/a)^2=3
, we need to find the value of a^6-1/a^6
. Let's start by solving the given equation.
(a+1/a)^2=3
Expanding the left-hand side of the equation, we get:
a^2 + 2a(1/a) + (1/a)^2 = 3
Simplifying the equation, we get:
a^2 + 2 + 1/a^2 = 3
Subtracting 2 from both sides, we get:
a^2 + 1/a^2 = 1
Simplifying the Equation
Multiplying both sides of the equation by a^2
, we get:
(a^2)^2 + a^2 = a^2
Simplifying further, we get:
a^4 + a^2 - a^2 = 0
Simplifying the equation, we get:
a^4 = 0
Finding the Value of a
Taking the fourth root of both sides, we get:
a = ±1
Finding the Value of a^6-1/a^6
Now that we have the value of a
, we can find the value of a^6-1/a^6
.
For a = 1
, we get:
a^6 - 1/a^6 = 1^6 - 1/1^6 = 1 - 1 = 0
For a = -1
, we get:
a^6 - 1/a^6 = (-1)^6 - 1/(-1)^6 = 1 - 1 = 0
Therefore, the value of a^6-1/a^6
is 0.