If 2 * 3 = 8, 3 * 4 = 81, then 4 * 5 = ?

Correct — A.
Rule: a * b = a raised to the power b.

Check on both given lines:
2 * 3 = 2³ = 8 ✓
3 * 4 = 3⁴ = 81 ✓

Apply it:
4 * 5 = 4⁵ = 4 × 4 × 4 × 4 × 4
= 16 × 16 × 4 = 1024 → option (a).

Why the others are wrong

• (b)
  1026 — 1026 is 4⁵ plus 2. The rule on both given lines is a bare power: 2³ = 8 and 3⁴ = 81 carry no added term.
• (c)
  256 — 256 is 4⁴, one power short. The second number is the exponent, so 4 * 5 means 4⁵, not 4 raised to 4.
• (d)
  2048 — 2048 is 2¹¹, twice the answer. 4⁵ = (2²)⁵ = 2¹⁰ = 1024, and doubling it adds a factor the rule never asked for.

Symbol-operation items redefine a sign such as * and give two worked lines. A rule counts only if it reproduces both lines exactly.

Try the simple candidates in turn: sum, product, squares, powers. The product gives 2 × 3 = 6 and 3 × 4 = 12, which fail at once. a to the power b gives 8 and 81, both exact, so it is the rule.

Key facts

• 2³ = 8, 3⁴ = 81 and 4⁵ = 1024.
• Powers of 2 worth knowing: 2⁸ = 256, 2¹⁰ = 1024, 2¹¹ = 2048.
• 4⁵ = (2²)⁵ = 2¹⁰.

Common traps

• Accepting a rule that fits the first line without testing the second
• Swapping base and exponent: 3² = 9, not 8, so the first line rules it out

No directly related past PYQ was found.