When adding normalized floating-point numbers 0.4546 × 10^5 and 0.5433 × 10^7, which alignment step is performed before addition?

Introduction / Context:
Floating-point addition requires aligning exponents so that the significands (mantissas) can be added. This step is core to understanding how CPUs implement IEEE-754 style arithmetic and why precision can be lost when magnitudes differ widely.

Given Data / Assumptions:

• Numbers: 0.4546 × 10^5 and 0.5433 × 10^7 (already normalized).
• Base-10 notation used for clarity; same idea applies in base-2.
• We want to perform addition, not multiplication.

Concept / Approach:

For addition, convert the number with the smaller exponent to the larger exponent by shifting the radix point (dividing its significand by the appropriate power of 10). This keeps relative value while making exponents equal.

Step-by-Step Solution:

Verification / Alternative check:

Converting the second to 10^5 would require moving the decimal right two places (54.33 × 10^5), which breaks normalized form and risks overflow in fixed-width mantissa; standard practice is to align the smaller exponent upward.

Why Other Options Are Wrong:

• No change: impossible—exponents differ.
• Changing 0.5433 × 10^7 to 54.33 × 10^5: de-normalizes the number unnecessarily.
• Forcing both to 10^6: not required and wastes precision.
• Changing to 10^9 arbitrarily: irrelevant to addition here.

Common Pitfalls:

Forgetting to shift the significand in the opposite direction to the exponent change; adding significands without alignment, leading to large error.

Final Answer:

0.4546 × 10^5 is changed to 0.004546 × 10^7 and 0.5433 × 10^7 is not changed