If {0.0015 * 10^(m)}{0.03 * 10^(k)} = 5 * 10^7, then m - k =

1. Translate the problem requirements: We have an equation with scientific notation where we need to find the difference between two exponents (\(\mathrm{m - k}\)). The equation shows a fraction equal to \(5 \times 10^7\), and we need to simplify the left side to match this form.
2. Simplify the numerical coefficients: Separate the decimal numbers from the powers of 10 and simplify the fraction of coefficients (\(0.0015 \div 0.03\)).
3. Combine the powers of 10: Use the rule that dividing powers means subtracting exponents, so \(10^\mathrm{m} \div 10^\mathrm{k} = 10^{(\mathrm{m-k})}\).
4. Match the simplified form to the given result: Compare our simplified expression to \(5 \times 10^7\) to determine the value of \((\mathrm{m-k})\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have here. We're given an equation that looks intimidating at first, but it's really just asking us to find the difference between two mystery numbers, m and k.

Think of it this way: we have a fraction on the left side that equals something specific on the right side (\(5 \times 10^7\)). The left side has two powers of 10 - one with exponent m in the numerator and one with exponent k in the denominator. When we simplify this fraction completely, we should get exactly \(5 \times 10^7\).

Our job is to find what \(\mathrm{m - k}\) equals. This is like solving a puzzle where we need to match the simplified left side to the right side.

Process Skill: TRANSLATE - Converting the complex-looking equation into a clear goal of finding m - k

2. Simplify the numerical coefficients

Now let's focus on just the decimal numbers first, ignoring the powers of 10 for a moment.

We need to calculate: \(0.0015 \div 0.03\)

To make this easier, let's think about this step by step:

- \(0.0015\) can be written as \(\frac{15}{10000}\)

- \(0.03\) can be written as \(\frac{3}{100}\)

When we divide fractions, we multiply by the reciprocal:

\(\frac{15}{10000} \div \frac{3}{100} = \frac{15}{10000} \times \frac{100}{3} = \frac{15 \times 100}{10000 \times 3} = \frac{1500}{30000} = \frac{1}{20} = 0.05\)

So the coefficient part simplifies to \(0.05\), which we can write as \(5 \times 10^{-2}\).

3. Combine the powers of 10

Now for the powers of 10 part. When we divide powers with the same base, we subtract the exponents.

So: \(10^\mathrm{m} \div 10^\mathrm{k} = 10^{(\mathrm{m-k})}\)

This is a fundamental rule: when dividing powers, subtract the bottom exponent from the top exponent.

Combining our results from steps 2 and 3:

Our left side becomes: \(5 \times 10^{-2} \times 10^{(\mathrm{m-k})} = 5 \times 10^{(\mathrm{m-k-2})}\)

Process Skill: SIMPLIFY - Breaking down complex expressions into manageable parts

4. Match the simplified form to the given result

Now we can set up our equation clearly:

\(5 \times 10^{(\mathrm{m-k-2})} = 5 \times 10^7\)

Since both sides have the same coefficient (5), the exponents must be equal:

\(\mathrm{m - k - 2} = 7\)

Solving for \(\mathrm{m - k}\):

\(\mathrm{m - k} = 7 + 2 = 9\)

Let's verify: if \(\mathrm{m - k} = 9\), then our left side becomes \(5 \times 10^{(9-2)} = 5 \times 10^7\) ✓

Final Answer

\(\mathrm{m - k} = 9\)

Checking against the answer choices, this matches choice (A) 9.

The key insight was recognizing that we could handle the decimal coefficients separately from the powers of 10, then combine everything systematically to match the given result.

Common Faltering Points

Errors while devising the approach

Students may panic when they see the complex-looking fraction with scientific notation and attempt to convert everything to decimal form first. This leads to unnecessary calculations with very small and large numbers (like \(0.0015\) and \(50,000,000\)), making the problem much more difficult than it needs to be. The key insight is recognizing that we can handle the decimal coefficients separately from the powers of 10.

Some students may try to solve for the individual values of m and k rather than recognizing that the question only asks for \(\mathrm{m - k}\). This misconception leads them down a more complex path, trying to set up multiple equations when a single approach focusing on the difference is sufficient.

Errors while executing the approach

When calculating \(0.0015 \div 0.03\), students often make mistakes with decimal placement. Common errors include getting \(0.5\) instead of \(0.05\), or miscounting zeros when converting to fractions. This error propagates through the entire solution and leads to an incorrect final answer.

Students may confuse the rule for dividing powers of 10. Instead of correctly applying \(10^\mathrm{m} \div 10^\mathrm{k} = 10^{(\mathrm{m-k})}\), they might add the exponents (\(10^{(\mathrm{m+k})}\)) or make other exponent rule errors. This is a fundamental algebraic mistake that completely derails the solution.

After finding that \(0.0015 \div 0.03 = 0.05 = 5 \times 10^{-2}\), students may incorrectly combine this with \(10^{(\mathrm{m-k})}\). They might write \(5 \times 10^{(\mathrm{m-k+2})}\) instead of the correct \(5 \times 10^{(\mathrm{m-k-2})}\), forgetting that adding a negative exponent is the same as subtracting.

Errors while selecting the answer

No likely faltering points - once students correctly solve the equation \(\mathrm{m - k - 2} = 7\) to get \(\mathrm{m - k} = 9\), the answer choice (A) 9 is straightforward to identify.