Scientific notation multiplication — evaluate the statement: “(5 × 10^6) × (4 × 10^3) = 20 × 10^18.” Is this mathematically correct?

Introduction / Context:Accurate manipulation of powers of ten is crucial in electronics (for example, converting microfarads to farads or kilohertz to hertz) and in data-sheet calculations. This question asks you to validate a multiplication written in scientific notation.

Given Data / Assumptions:

• Numbers: 5 × 10^6 and 4 × 10^3.
• Standard rules for scientific notation apply.
• No units are required; evaluation is purely numerical.

Concept / Approach:When multiplying numbers in scientific notation, multiply the mantissas and add the exponents: (a × 10^m) × (b × 10^n) = (a*b) × 10^(m+n). Afterward, normalize the mantissa if necessary so that it lies between 1 and 10, adjusting the exponent accordingly.

Step-by-Step Solution:

Verification / Alternative check:Compute in plain numbers: 5,000,000 × 4,000 = 20,000,000,000 = 2.0 × 10^10. The statement “20 × 10^18” is off by nine orders of magnitude in the exponent.

Why Other Options Are Wrong:

• Correct: Incorrect because the exponents must be added (6 + 3 = 9), not multiplied or concatenated to 18.
• Correct only if significant figures are ignored: Significant figures do not change the exponent arithmetic.
• Cannot be decided without units: Units are irrelevant to the numeric operation.

Common Pitfalls:Confusing exponent addition with multiplication, or forgetting to normalize mantissas. Always multiply coefficients first, then add exponents and normalize.

Final Answer:Incorrect — the correct product is 2.0 × 10^10, not 20 × 10^18.