The product 54.327 × 357.2 × 0.0057 is to be rewritten by shifting decimal points only. Which of the following rewritten products is exactly equal to this value?

Introduction / Context:
This question focuses on handling decimal numbers by shifting decimal points while keeping the overall product unchanged. It tests your understanding that multiplying or dividing by 10, 100, and similar powers of 10 can be balanced across factors so that the final result remains the same.

Given Data / Assumptions:

• Original product: 54.327 × 357.2 × 0.0057.
• We must choose an equivalent product formed by shifting decimal points.
• No rounding or change in the actual numerical value is allowed.

Concept / Approach:
Shifting a decimal point to the left divides a number by a power of 10, and shifting it to the right multiplies it by a power of 10. Any decrease in one factor can be compensated by a corresponding increase in another factor so that the total product remains constant. We can compare numerically or, more simply, express the decimals using powers of 10.

Step-by-Step Solution:
Step 1: Compute or inspect the original product numerically for reference. Step 2: Consider option A: 5.4327 × 3.572 × 5.7. Step 3: Note that 54.327 = 5.4327 × 10 and 357.2 = 3.572 × 100, while 0.0057 = 5.7 / 1000. Step 4: The combined power of 10 in the original product is 10 × 100 × (1 / 1000) = 1, so there is no net effect from the powers of 10. Step 5: Therefore, 54.327 × 357.2 × 0.0057 = (5.4327 × 10) × (3.572 × 100) × (5.7 / 1000). Step 6: Rearranging, this equals 5.4327 × 3.572 × 5.7 × (10 × 100 / 1000) = 5.4327 × 3.572 × 5.7. Step 7: Thus, option A is exactly equal to the original product.

Verification / Alternative check:
You may calculate the numerical value of 54.327 × 357.2 × 0.0057 and 5.4327 × 3.572 × 5.7 using a calculator. Both will yield the same decimal value (approximately 110.61194508), confirming the equality.

Why Other Options Are Wrong:
Option B uses 0.57 instead of 5.7, which introduces an extra division by 10, making the product ten times smaller.
Option C uses 0.0000057 along with large integers, resulting in an overall product that is ten times larger than the original.
Option E shifts decimal points in a way that changes the net power of 10 and therefore does not preserve the product.
Option D "None of these" is incorrect because option A is a valid equivalent expression.

Common Pitfalls:
A common mistake is to shift decimal points in several numbers without checking the combined effect of powers of 10. It is important to track how many times you effectively multiply or divide by 10 overall. Another pitfall is relying only on approximate mental calculations instead of checking powers of 10 systematically.

Final Answer:
The expression that is exactly equal to the original product is 5.4327 x 3.572 x 5.7.