(a+1/a)^2=3 Then A^6-1/a^6

Suzhoumeite

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Jul 03, 2024

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(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.