Given 2ⁿ = 2^(x²) × 16ˣ × 8 with x 0, find x as an expression in n and state any restrictions on n.

Question

Given 2ⁿ = 2^(x²) × 16ˣ × 8 with x > 0, find x as an expression in n and state any restrictions on n.

Accepted Answer

Write every factor as a power of 2: 16ˣ = (2⁴)ˣ = 2^(4x) and 8 = 2³ 2ⁿ = 2^(x²) × 2^(4x) × 2³ Add the indices: 2ⁿ = 2^(x² + 4x + 3) The bases are equal, so n = x² + 4x + 3 Rearrange: x² + 4x + (3 − n) = 0 x = [-4 ± √(4² − 4 × 1 × (3 − n))] ÷ 2 x = [-4 ± √(16 − 12 + 4n)] ÷ 2 x = [-4 ± √(4n + 4)] ÷ 2 x = [-4 ± 2√(n + 1)] ÷ 2 x = −2 ± √(n + 1) x > 0, so take the positive square root: x = −2 + √(n + 1) x > 0 needs √(n + 1) > 2, so n + 1 > 4 x = −2 + √(n + 1) and n > 3 Answer: x = −2 + √(n + 1) and n > 3

4MA1/1HR Jun 2023, Question 24: Exponential Equations

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