If 16 * 2$^{(x + 1)}$ = 4$^x$ * 8$^{(1 - x)}$, find the value of x.

Analyze the question: We have an exponential equation with different bases (16, 4, 8, 2). The trick is to write everything as a power of 2, then equate the indices.

Concept: If aᵐ = aⁿ, then m = n. Also, aᵐ × aⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ.

Solution:

Rewrite each number as a power of 2:

• 16 = 2⁴
• 4 = 2², so 4ˣ = 2²ˣ
• 8 = 2³, so 8^(1−x) = 2^(3(1−x)) = 2^(3−3x)

Left side: 2⁴ × 2^(x+1) = 2^(4 + x + 1) = 2^(x+5)

Right side: 2^(2x) × 2^(3−3x) = 2^(2x + 3 − 3x) = 2^(3−x)

Equate the indices: x + 5 = 3 − x 2x = 3 − 5 2x = −2 x = −1

Check: Left: 16 × 2^(−1+1) = 16 × 1 = 16 Right: 4^(−1) × 8^(2) = (1/4) × 64 = 16 ✓

Answer: x = −1, which is option D.

Tip: In any exponential equation with different bases, first convert everything to the same prime base (2, 3, 5...). Then just equate the powers. Always substitute your answer back to confirm it.