If x = 1/2^2 * 3^2 * 4^2 * 5^2 is expressed as a decimal, how many distinct nonzero digits...

1. Translate the problem requirements: We need to express \(\mathrm{x = \frac{1}{2^2 \times 3^2 \times 4^2 \times 5^2}}\) as a decimal and count how many different nonzero digits appear in that decimal representation.
2. Simplify the denominator by factoring: Break down all terms to prime factors to see the true structure of the fraction.
3. Convert to decimal form: Transform the simplified fraction into its decimal representation.
4. Identify distinct nonzero digits: Examine the decimal to count how many different nonzero digits appear.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to do. We have a fraction \(\mathrm{x = \frac{1}{2^2 \times 3^2 \times 4^2 \times 5^2}}\) and we need to convert this to a decimal number. Once we have that decimal, we need to count how many different nonzero digits appear in it.

Think of it like this: if our decimal turned out to be 0.123400, the distinct nonzero digits would be 1, 2, and 3 (we don't count 4 twice, and we ignore the zeros). So we'd have 3 distinct nonzero digits.

Process Skill: TRANSLATE

2. Simplify the denominator by factoring

Now let's break down our denominator step by step. We have \(\mathrm{2^2 \times 3^2 \times 4^2 \times 5^2}\), but notice that 4 is not a prime number - it's actually \(\mathrm{2^2}\).

So \(\mathrm{4^2 = (2^2)^2 = 2^4}\)

This means our denominator becomes:
\(\mathrm{2^2 \times 3^2 \times 2^4 \times 5^2}\)

When we multiply powers of the same base, we add the exponents:
\(\mathrm{2^2 \times 2^4 = 2^6}\)

So our denominator simplifies to:
\(\mathrm{2^6 \times 3^2 \times 5^2}\)

Let's calculate this step by step:

• \(\mathrm{2^6 = 64}\)
• \(\mathrm{3^2 = 9}\)
• \(\mathrm{5^2 = 25}\)

So our denominator = \(\mathrm{64 \times 9 \times 25 = 64 \times 225 = 14,400}\)

Therefore: \(\mathrm{x = \frac{1}{14,400}}\)

Process Skill: SIMPLIFY

3. Convert to decimal form

Now we need to convert \(\mathrm{\frac{1}{14,400}}\) to a decimal. Let's do this division:

\(\mathrm{1 \div 14,400 = 0.0000694444...}\)

To be more precise, let's work this out systematically:

• \(\mathrm{1.000000... \div 14,400}\)
• This gives us \(\mathrm{0.0000694444...}\) (the 4 repeats)

We can verify this makes sense because 14,400 is a relatively large number, so 1 divided by it should give us a small decimal that starts with several zeros.

Actually, let's be more careful with this calculation:
\(\mathrm{\frac{1}{14,400} = \frac{1}{144 \times 100} = \frac{1}{144} \div 100}\)

\(\mathrm{\frac{1}{144} = \frac{1}{12^2} = \frac{1}{144} \approx 0.006944...}\)
So \(\mathrm{\frac{1}{14,400} = 0.00006944...}\)

More precisely: \(\mathrm{x = 0.00006944444...}\) (where the 4 repeats infinitely)

4. Identify distinct nonzero digits

Looking at our decimal \(\mathrm{0.00006944444...}\), let's identify all the distinct nonzero digits:

• The digit 6 appears once
• The digit 9 appears once
• The digit 4 appears multiple times (repeating)

So the distinct nonzero digits are: 6, 9, and 4

That gives us exactly 3 distinct nonzero digits.

4. Final Answer

The decimal representation of \(\mathrm{x = \frac{1}{2^2 \times 3^2 \times 4^2 \times 5^2}}\) has exactly 3 distinct nonzero digits: 6, 9, and 4.

The answer is C. Three.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "distinct nonzero digits" means

Students may confuse "distinct nonzero digits" with "total number of nonzero digits" or "total number of digits." For example, if the decimal is \(\mathrm{0.00006944444...}\), they might count every occurrence of the digit 4 separately instead of counting it just once as a distinct digit. The question asks for how many different nonzero digits appear, not how many times they appear.

2. Incorrectly handling the denominator factorization

Students often miss that \(\mathrm{4 = 2^2}\), so \(\mathrm{4^2 = 2^4}\). They may treat the denominator as \(\mathrm{2^2 \times 3^2 \times 4^2 \times 5^2}\) without recognizing that this contains repeated prime factors. This leads to an incorrect denominator value and consequently an incorrect decimal representation.

Errors while executing the approach

1. Arithmetic errors in calculating the denominator

When computing \(\mathrm{2^6 \times 3^2 \times 5^2 = 64 \times 9 \times 25}\), students frequently make multiplication errors. Common mistakes include calculating \(\mathrm{64 \times 9 = 576}\) incorrectly, or getting the final product \(\mathrm{64 \times 9 \times 25 = 14,400}\) wrong, leading to division by the wrong number.

2. Imprecise decimal conversion

Students may perform the division \(\mathrm{1 \div 14,400}\) carelessly, either stopping the calculation too early or making errors in the long division process. This can result in an incorrect decimal like \(\mathrm{0.0000694}\) instead of \(\mathrm{0.00006944444...}\), missing the repeating pattern of 4s that affects the final count of distinct digits.

3. Misidentifying the decimal digits

Even with the correct decimal \(\mathrm{0.00006944444...}\), students may misread the digits, especially confusing 6 and 9, or may incorrectly include or exclude certain digits when identifying the nonzero ones.

Errors while selecting the answer

1. Counting zeros as distinct digits

After obtaining the decimal \(\mathrm{0.00006944444...}\), students might count the zeros as distinct digits, leading them to count more digits than the actual nonzero distinct digits (6, 9, 4). This would lead them to select a higher answer choice than C.

2. Double-counting repeated digits

Students may count the digit 4 multiple times since it repeats in the decimal representation, arriving at more than 3 distinct digits. This conceptual error about what "distinct" means can lead them to choose answer choice D (Seven) or E (Ten) instead of the correct answer C (Three).