If (a + 1/a) = 7√3, then what is the value of (a^6 + a^(-6))?

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Accepted Answer

Given: a + 1/a = 7√3

Step 1: Find (a^2 + 1/a^2)
Using the identity:
(a + 1/a)^2 = a^2 + 2 + 1/a^2
=> a^2 + 1/a^2 = (a + 1/a)^2 - 2
= (7√3)^2 - 2
= 49 * 3 - 2
= 147 - 2 = 145

Step 2: Find (a^3 + 1/a^3)
Using the identity:
(a + 1/a)^3 = a^3 + 3a + 3(1/a) + 1/a^3
=> a^3 + 1/a^3 = (a + 1/a)^3 - 3(a + 1/a)
= (7√3)^3 - 3 * 7√3
= 7^3 * (√3)^3 - 21√3
= 343 * 3√3 - 21√3
= (1029√3) - 21√3 = 1008√3

Step 3: Find (a^6 + 1/a^6)
Using the identity:
(a^3 + 1/a^3)^2 = a^6 + 2 + 1/a^6
=> a^6 + 1/a^6 = (a^3 + 1/a^3)^2 - 2
= (1008√3)^2 - 2
= 1008^2 * 3 - 2
= (1,016,064) * 3 - 2
= 3,048,192 - 2 = 3,048,190

Therefore, the value of (a^6 + a^(-6)) is 3,048,190.

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