If x⁴ + 1/x⁴ = 14, then the value of x³ + 1/x³ is: A) 3√6 B) 18/√6 C) 9√(2/3)…

For a real number x, power-sums of the form xn + 1/xn are linked across exponents by two standard identities: squaring links n to 2n via (xn + 1/xn)2 = x2n + 1/x2n + 2, and the cube identity (x + 1/x)3 = x3 + 1/x3 + 3(x + 1/x) links x + 1/x directly to x3 + 1/x3.

Application of these identities to this item:

1. Use the squaring identity with n = 2: (x2 + 1/x2)2 = x4 + 1/x4 + 2 = 14 + 2 = 16, so x2 + 1/x2 = 4 (positive root, since x2 + 1/x2 > 0 for every nonzero real x).

2. Use the squaring identity with n = 1: (x + 1/x)2 = x2 + 1/x2 + 2 = 4 + 2 = 6, so x + 1/x = ±√6 — the sign matches the sign of x itself (x + 1/x > 0 for x > 0, and < 0 for x < 0). Every form offered in the stem (A–D) is a positive surd, so this item is read on the x > 0 branch: x + 1/x = √6. (On the x < 0 branch, x + 1/x = −√6 and the working below would instead give x3 + 1/x3 = −3√6, a value none of the offered forms represent.)

3. Apply the cube identity: (x + 1/x)3 = x3 + 1/x3 + 3(x + 1/x). Substituting: (√6)3 = x3 + 1/x3 + 3√6, i.e. 6√6 = x3 + 1/x3 + 3√6, so x3 + 1/x3 = 3√6.

Cross-check — simplify each stem expression independently and compare it to 3√6:

• 18/√6 = 18√6/6 = 3√6 (rationalising the denominator) — matches.

• 9√(2/3) = 9√2/√3 = 9√6/3 = 3√6 (rationalising) — matches.

• 3√2 stays 3√2, a surd of 2, not of 6 — does not match (squaring confirms: (3√2)2 = 18, while (3√6)2 = 54).

So x3 + 1/x3 = 3√6, which is exactly the value taken by the forms labelled A, B and C in the stem; the form labelled D is a different value.