If a³ + 1/a³ = 2 (a 0), then find the value of a + 1/a.

Question

If a³ + 1/a³ = 2 (a > 0), then find the value of a + 1/a.

Accepted Answer

Given: a³ + 1/a³ = 2 and a > 0.

We know the identity:
(a + 1/a)³ = a³ + 1/a³ + 3(a + 1/a)

Let x = a + 1/a.
Then,
x³ = a³ + 1/a³ + 3x

Substitute the given value:
2 = a³ + 1/a³

So,
x³ = 2 + 3x

Rearranging:
x³ - 3x - 2 = 0

Try possible integer roots (factors of 2): ±1, ±2

For x = 1:
1 - 3 - 2 = -4 ≠ 0

For x = 2:
8 - 6 - 2 = 0

So, x = 2 is a root.

Since a > 0, a + 1/a = 2.

Therefore, the value of a + 1/a is 2.

Answer

Answer

Answer