If r and s are positive integers such that \((2^{\mathrm{r}})(4^{\mathrm{s}}) = 16\), then 2r + s =

1. Translate the problem requirements: We need to find the value of \(2\mathrm{r} + \mathrm{s}\), where r and s are positive integers that satisfy the equation \((2^\mathrm{r})(4^\mathrm{s}) = 16\)
2. Express everything in terms of the same base: Convert \(4^\mathrm{s}\) and 16 to powers of 2 so we can work with a single base throughout
3. Apply the fundamental exponent rule: Use the property that when multiplying powers with the same base, we add the exponents
4. Equate exponents to solve for the relationship: Since we have equal expressions with the same base, their exponents must be equal
5. Find values and calculate the target expression: Determine r and s, then compute \(2\mathrm{r} + \mathrm{s}\)

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for. We have an equation \((2^\mathrm{r})(4^\mathrm{s}) = 16\), where r and s are positive integers. Our goal is to find the value of \(2\mathrm{r} + \mathrm{s}\).

Think of this like a puzzle where we need to figure out what values of r and s make this equation true, and then use those values to calculate our final answer.

Process Skill: TRANSLATE - Converting the problem statement into a clear mathematical goal

2. Express everything in terms of the same base

Here's the key insight: we want to work with the same base throughout to make our calculations easier. Since we see powers of 2 and 4, and 16 is also a power of 2, let's convert everything to base 2.

Let's think about this step by step:

• 4 can be written as \(2^2\) (since \(2 \times 2 = 4\))
• So \(4^\mathrm{s}\) becomes \((2^2)^\mathrm{s}\)
• 16 can be written as \(2^4\) (since \(2 \times 2 \times 2 \times 2 = 16\))

Now our equation becomes:
\((2^\mathrm{r}) \times (2^2)^\mathrm{s} = 2^4\)

3. Apply the fundamental exponent rule

When we have \((2^2)^\mathrm{s}\), we can simplify this using the power rule. When you raise a power to another power, you multiply the exponents:
\((2^2)^\mathrm{s} = 2^{(2\mathrm{s})}\)

So our equation is now:
\((2^\mathrm{r}) \times (2^{2\mathrm{s}}) = 2^4\)

When we multiply powers with the same base, we add the exponents:
\(2^{(\mathrm{r} + 2\mathrm{s})} = 2^4\)

4. Equate exponents to solve for the relationship

Here's a fundamental principle: if two expressions with the same base are equal, then their exponents must be equal too.

Since \(2^{(\mathrm{r} + 2\mathrm{s})} = 2^4\), we know that:
\(\mathrm{r} + 2\mathrm{s} = 4\)

Now we need to find positive integer values for r and s that satisfy this equation.

Process Skill: CONSIDER ALL CASES - We need to check which positive integer combinations work

Let's try different positive values:

• If \(\mathrm{s} = 1\): \(\mathrm{r} + 2(1) = 4\), so \(\mathrm{r} = 2\)
• If \(\mathrm{s} = 2\): \(\mathrm{r} + 2(2) = 4\), so \(\mathrm{r} = 0\) (but r must be positive, so this doesn't work)
• If \(\mathrm{s} = 3\) or higher: r would be negative (doesn't work)

So the only solution is \(\mathrm{r} = 2\) and \(\mathrm{s} = 1\).

5. Find values and calculate the target expression

Let's verify our solution works:

• \(\mathrm{r} = 2, \mathrm{s} = 1\)
• \((2^2)(4^1) = 4 \times 4 = 16\) ✓

Now we can calculate \(2\mathrm{r} + \mathrm{s}\):
\(2\mathrm{r} + \mathrm{s} = 2(2) + 1 = 4 + 1 = 5\)

Final Answer

The value of \(2\mathrm{r} + \mathrm{s} = 5\), which corresponds to answer choice (D).

Common Faltering Points

Errors while devising the approach

1. Not recognizing the need to use the same base throughout
Many students get stuck because they try to work with the equation \((2^\mathrm{r})(4^\mathrm{s}) = 16\) without converting everything to the same base. They might attempt to solve this by trying different combinations of r and s through trial and error, which is time-consuming and error-prone. The key insight is recognizing that since all terms can be expressed as powers of 2, converting to base 2 makes the problem much more manageable.

2. Forgetting to consider the constraint that r and s must be positive integers
Students often miss or overlook the specific constraint that r and s are positive integers. This constraint is crucial because it limits the possible solutions. Without paying attention to this requirement, students might accept solutions where r or s equals zero or negative values, leading to incorrect final answers.

Errors while executing the approach

1. Incorrectly applying exponent rules
When converting \(4^\mathrm{s}\) to base 2, students commonly make errors with the power rule. They might write \(4^\mathrm{s} = 2^{2\mathrm{s}}\) incorrectly as \(2^{(2+\mathrm{s})}\) instead of \(2^{(2\mathrm{s})}\). This is a critical error because \((2^2)^\mathrm{s} = 2^{(2\times\mathrm{s})} = 2^{(2\mathrm{s})}\), not \(2^{(2+\mathrm{s})}\). This mistake leads to the wrong equation and ultimately the wrong answer.

2. Making arithmetic errors when checking solutions
When verifying that \(\mathrm{r} = 2\) and \(\mathrm{s} = 1\) work in the original equation, students might make simple calculation mistakes. For example, they might incorrectly calculate \((2^2)(4^1)\) as something other than 16, or make errors when substituting back into the constraint equation \(\mathrm{r} + 2\mathrm{s} = 4\).

Errors while selecting the answer

1. Calculating the wrong expression for the final answer
Even after correctly finding \(\mathrm{r} = 2\) and \(\mathrm{s} = 1\), students might calculate the wrong final expression. They might compute \(\mathrm{r} + \mathrm{s} = 3\) instead of \(2\mathrm{r} + \mathrm{s} = 5\), or make other substitution errors. Since the question asks specifically for \(2\mathrm{r} + \mathrm{s}\), students must be careful to calculate \(2(2) + 1 = 5\), not just \(\mathrm{r} + \mathrm{s}\) or some other combination of r and s.