Multiplying numbers in scientific notation Compute (12.6 × 10^6) * (10.5 × 10^3) and express the result in proper scientific notation (mantissa between 1 and 10).

Introduction / Context:
Scientific notation accelerates multiplication and division by separating magnitude (10^n) from the significant digits (mantissa). This is particularly helpful in electronics when combining values across very different scales, such as gains, frequencies, or time constants.

Given Data / Assumptions:

• Numbers: 12.6 × 10^6 and 10.5 × 10^3.
• We seek normalized scientific notation (1 ≤ mantissa < 10).
• Arithmetic is exact to at least three significant digits.

Concept / Approach:
Multiply mantissas and add exponents: (a × 10^m) * (b × 10^n) = (a*b) × 10^(m+n). If the mantissa product is not between 1 and 10, re-normalize by adjusting the exponent accordingly. Keep an appropriate number of significant figures based on inputs (three sig figs here are reasonable: 12.6 and 10.5 each have three).

Step-by-Step Solution:

Verification / Alternative check:
Use a rounded mantissa to three significant digits consistent with inputs: 1.323 × 10^11 ≈ 1.32 × 10^11. A calculator check confirms the numeric value: 12.6e6 * 10.5e3 = 1.323e11, matching the normalized result.

Why Other Options Are Wrong:

Common Pitfalls:
Forgetting to renormalize the mantissa; adding exponents incorrectly; rounding too early, which can introduce noticeable errors when cascaded across multiple steps.

Final Answer:
1.32 × 10^11