Which of the following rewritten products is exactly equal to 54.327 × 357.2 × 0.0057?

Introduction / Context:
This question tests your ability to manipulate decimal numbers by shifting decimal points and compensating with powers of 10. It is a typical decimal fraction question where you must recognize that moving the decimal point changes the magnitude of a number and that equivalent products can be formed by balancing those shifts across all factors.

Given Data / Assumptions:

• Original product: 54.327 × 357.2 × 0.0057.
• Several alternative products are provided, each formed by moving decimal points.
• We must find which alternative product has the same numerical value as the original expression.
• No approximation is allowed; the equality must be exact.

Concept / Approach:
When you move the decimal point to the left, you are effectively dividing the number by 10 for each place shifted. Moving it to the right multiplies it by 10. If you change one factor by a power of 10, you must adjust another factor by an opposite power of 10 to keep the overall product unchanged. The core idea is to track how many decimal places have been shifted and translate that into powers of 10.

Step-by-Step Solution:
Start with 54.327. Moving the decimal one place left gives 5.4327, which is 54.327 ÷ 10.Next consider 357.2. Moving the decimal two places left gives 3.572, which is 357.2 ÷ 100.Together, these two changes divide the original product by 10 × 100 = 1000.To keep the product unchanged, we must multiply the third factor 0.0057 by 1000.0.0057 × 1000 = 5.7.So an equivalent product is 5.4327 × 3.572 × 5.7.Therefore, option 5.4327 × 3.572 × 5.7 is exactly equal to the original expression.

Verification / Alternative check:
We can verify numerically. Compute the original product and the candidate product. Both 54.327 × 357.2 × 0.0057 and 5.4327 × 3.572 × 5.7 evaluate to the same decimal value when calculated carefully. This confirms that our reasoning with powers of 10 and decimal shifts is accurate.

Why Other Options Are Wrong:
Option 5.4327 × 3.572 × 0.57 only multiplies the last factor by 100, while the first two factors together divided the product by 1000, so the overall product becomes one tenth of the original. Option 54327 × 3572 × 0.0000057 changes factors by far larger powers of 10 in an unbalanced way, not preserving equality. Option 54.327 × 3.572 × 0.57 changes only some factors and again does not balance the net power of 10. Therefore, these are not numerically equal to the original product.

Common Pitfalls:
Students sometimes move decimal points without explicitly thinking in terms of powers of 10, which makes it easy to lose track of how much the product has changed. Another common mistake is to assume that because numbers look similar, the expressions must be equal, without checking the net number of decimal shifts. Always count the total power of 10 applied across all factors and ensure that the net effect is zero if you want an equivalent product.

Final Answer:
The expression equivalent to 54.327 × 357.2 × 0.0057 is 5.4327 × 3.572 × 5.7.