If (a + 1/a) = 7√3, then what is the value of (a⁶ + a⁻⁶)?
3048190
Given:
If (a + 1/a) = 7√3 , then we need to find the value of (a⁶ + a⁻⁶).
Step 1: Square both sides of the given equation:
(a + 1/a)² = (7√3)²
Expanding the left side:
a² + 2 + 1/a² = 49 × 3
So, we get:
a² + 1/a² + 2 = 147
Subtract 2 from both sides:
a² + 1/a² = 145
Step 2: Cube both sides of the equation (a + 1/a):
(a + 1/a)³ = (7√3)³
Expanding the left side:
a³ + 1/a³ + 3(a + 1/a) = (7√3)³
Since (a + 1/a) = 7√3, substitute and simplify:
a³ + 1/a³ + 3 × 7√3 = 7³ × (√3)³a³ + 1/a³ + 21√3 = 1029
Step 3: Solve for a³ + 1/a³:
a³ + 1/a³ = 1029 - 21√3
Step 4: Compute (a⁶ + a⁻⁶):
a⁶ + a⁻⁶ = (a³ + 1/a³)² - 2a⁶ + a⁻⁶ = (1029 - 21√3)² - 2
Conclusion:
The value of (a⁶ + a⁻⁶) is 3048190.
Correct Answer:
The correct answer is: Option 3: 3048190.
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