If d = 1/2^3 * 5^7 is expressed as a terminating decimal, how many nonzero digits will d have?

1. Translate the problem requirements: We need to convert the fraction \(\mathrm{d = \frac{1}{2^3 \times 5^7}}\) into its decimal form and count how many digits in that decimal are nonzero
2. Convert to standard decimal form: Manipulate the fraction to get a power of 10 in the denominator, since decimals are based on powers of 10
3. Express as decimal and count nonzero digits: Write out the decimal representation and identify which digit positions contain nonzero values

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 \(\mathrm{d = \frac{1}{2^3 \times 5^7}}\), and we need to convert this fraction into a decimal number. Then we need to count how many of the digits in that decimal are NOT zero.

First, let's calculate what's in the denominator. We have \(\mathrm{2^3 = 2 \times 2 \times 2 = 8}\), and \(\mathrm{5^7 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 78,125}\). So our fraction is \(\mathrm{d = \frac{1}{8 \times 78,125} = \frac{1}{625,000}}\).

Now we need to convert \(\mathrm{\frac{1}{625,000}}\) to a decimal and see how many nonzero digits it contains.

Process Skill: TRANSLATE - Converting the mathematical expression into concrete numbers we can work with

2. Convert to standard decimal form

Here's the key insight: to convert a fraction to a decimal easily, we want the denominator to be a power of 10 (like 10, 100, 1,000, etc.) because that makes it simple to place the decimal point.

We currently have \(\mathrm{d = \frac{1}{2^3 \times 5^7}}\). Let's think about what we need to make this denominator into a power of 10. Powers of 10 always look like \(\mathrm{10^n = (2 \times 5)^n = 2^n \times 5^n}\).

Looking at our denominator \(\mathrm{2^3 \times 5^7}\), we have 3 factors of 2 and 7 factors of 5. To make a power of 10, we need equal numbers of 2's and 5's. Since we have more 5's than 2's, we need to add more 2's.

To get equal powers, we need 7 factors of both 2 and 5. We already have \(\mathrm{2^3 \times 5^7}\), so we need 4 more factors of 2. We can do this by multiplying both numerator and denominator by \(\mathrm{2^4 = 16}\):

\(\mathrm{d = \frac{1}{2^3 \times 5^7} \times \frac{2^4}{2^4} = \frac{2^4}{2^3 \times 2^4 \times 5^7} = \frac{16}{2^7 \times 5^7} = \frac{16}{10^7}}\)

Now we have \(\mathrm{\frac{16}{10,000,000}}\), which is much easier to convert to decimal form!

3. Express as decimal and count nonzero digits

Now that we have \(\mathrm{d = \frac{16}{10^7} = \frac{16}{10,000,000}}\), converting to decimal is straightforward.

When we divide 16 by 10,000,000, we're essentially taking the number 16 and moving the decimal point 7 places to the left:
\(\mathrm{16 \div 10,000,000 = 0.0000016}\)

Let's verify this makes sense: \(\mathrm{0.0000016 \times 10,000,000 = 16}\) ✓

Now we need to count the nonzero digits in 0.0000016:
- The digit 1 is nonzero
- The digit 6 is nonzero
- All the zeros are... well, zero!

So we have exactly 2 nonzero digits: the 1 and the 6.

4. Final Answer

The decimal representation of \(\mathrm{d = \frac{1}{2^3 \times 5^7}}\) is 0.0000016, which contains exactly 2 nonzero digits.

The answer is (B) Two.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "nonzero digits" means: Students might count ALL digits in the decimal (including zeros) instead of counting only the digits that are not zero. This fundamental misunderstanding would lead them to count positions rather than actual nonzero values.

2. Not recognizing the need to convert to power of 10: Students might try to directly divide 1 by \(\mathrm{(2^3 \times 5^7)}\) using long division or calculator methods instead of recognizing that multiplying by appropriate powers of 2 or 5 can create a power of 10 in the denominator, making decimal conversion much cleaner.

3. Confusion about terminating decimal conditions: Students might not understand why this fraction produces a terminating decimal in the first place, or might waste time checking if it terminates when the problem already states it does.

Errors while executing the approach

1. Arithmetic errors in calculating powers: Students might miscalculate \(\mathrm{2^3 = 8}\), or more likely make errors in computing \(\mathrm{5^7 = 78,125}\), leading to an incorrect denominator value and subsequent wrong decimal conversion.

2. Incorrect decimal point placement: When converting \(\mathrm{\frac{16}{10^7}}\) to decimal form, students might miscount the number of places to move the decimal point, resulting in 0.000016 (6 zeros) or 0.00000016 (8 zeros) instead of the correct 0.0000016 (5 zeros).

3. Wrong multiplication to create equal powers: Students might multiply by \(\mathrm{5^4}\) instead of \(\mathrm{2^4}\), or miscalculate how many additional factors of 2 are needed to balance the \(\mathrm{2^3 \times 5^7}\) expression, leading to an incorrect form for easy decimal conversion.

Errors while selecting the answer

1. Counting zeros instead of nonzeros: After correctly finding 0.0000016, students might count the 5 zeros and select answer choice (E) Ten or some other incorrect option, rather than counting the 2 nonzero digits (1 and 6).

2. Including decimal places in the count: Students might count the decimal point itself as a "digit" or include leading zeros in their nonzero digit count, leading to answers like 3 or 7 instead of the correct 2.