The decimal expansion of the rational number 17/(2^2 × 5) will terminate after how many places of decimal? (A) 1 (B) 2 (C) 3 (D) 4

Understand the Problem

The question is asking about the decimal expansion of a rational number, specifically relating to how many decimal places it will terminate after when expressed in decimal form. This involves understanding the properties of rational numbers and their factors.

Answer

The decimal expansion will terminate after 3 decimal places.

Steps to Solve

Understand the rational number structure
The number given is ( \frac{17}{2^2 \times 5} ). Rational numbers terminate in decimal form when their denominator (in simplest form) has only the prime factors 2 and 5.
Simplify the denominator
The denominator is ( 2^2 \times 5 ). This indicates that the factors are already 2 and 5, which is suitable for terminating decimals.
Count the powers of each prime
The powers of the primes in the denominator are:

For 2: ( 2^2 ) (which contributes 2 decimal places)
For 5: ( 5^1 ) (which contributes 1 decimal place)

Determine the maximum decimal places
The number of decimal places is determined by the largest power among the factors of 2 and 5.

Here, the maximum between 2 and 1 is 2. Thus, the terminating decimal will have a total of ( 2 + 1 = 3 ) decimal places.

The decimal expansion of the rational number ( \frac{17}{2^2 \times 5} ) will terminate after 3 decimal places.

More Information

Rational numbers can either terminate or repeat in decimal form. The main criterion for a rational number to have a terminating decimal is that the denominator, once simplified, must only include the prime factors 2 and 5.

Tips

Forgetting to simplify the rational number before analyzing the factors can lead to incorrect decimal place counts.
Not correctly identifying the prime factors may result in miscalculating the decimal places.

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