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Question:

If x³ + y³ = 416 and x + y = 8, then find x⁴ + y⁴.

Explanation:

We know, (x + y)³ = x³ + y³ + 3xy(x + y)
⇒ 8³ = 416 + 3 × xy × 8
⇒ 512 - 416 = 24xy
⇒ xy = 96/24 = 4

Now, x² + y² = (x + y)² - 2xy = 64 - 8 = 56

Squaring both sides we get,

(x² + y²)² = 56²
⇒ x⁴ + y⁴ + 2x²y² = 3136
⇒ x⁴ + y⁴ + 32 = 3136
⇒ x⁴ + y⁴ = 3104

Hence, option (d).