If x⁴ + 1/x⁴ = 527, then the value of x + 1/x is

Concept

For any non-zero number, squaring the sum of a quantity and its reciprocal links consecutive powers: (a + 1/a)2 = a2 + 1/a2 + 2. So adding 2 to a “sum + reciprocal of the square” gives a perfect square, and taking its square root steps the power down by half each time.

Application

1. Start from the given: x4 + 1/x4 = 527.

2. Use (x2 + 1/x2)2 = x4 + 1/x4 + 2 = 527 + 2 = 529.

3. Take the positive square root: x2 + 1/x2 = √529 = 23.

4. Use (x + 1/x)2 = x2 + 1/x2 + 2 = 23 + 2 = 25.

5. Take the positive square root: x + 1/x = √25 = 5.

Cross-check

Work forward from 5: if x + 1/x = 5, then x2 + 1/x2 = 52 − 2 = 23, and x4 + 1/x4 = 232 − 2 = 527, which matches the given value, so x + 1/x = 5.

Note on sign: algebraically (x + 1/x)2 = 25 gives x + 1/x = ±5, since x + 1/x = −5 also yields x4 + 1/x4 = 527. Among the values offered, only the positive value 5 appears, which is the standard convention (x > 0) for this question, so x + 1/x = 5.