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 \(2r + s\), where r and s are positive integers that satisfy the equation \((2^r)(4^s) = 16\)
2. Express everything as powers of the same base: Convert \(4^s\) and 16 to powers of 2 so we can work with a single base throughout the equation
3. Apply the fundamental exponent rule: Use the property that when multiplying powers with the same base, we add the exponents to simplify the left side
4. Set up the equation for direct comparison: Since both sides are now powers of 2, we can equate the exponents directly
5. Solve systematically for positive integer solutions: Find values of r and s that satisfy our exponent equation, then calculate \(2r + s\)

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for. We have an equation with two unknown positive integers, r and s, and we need to find the value of the expression \(2r + s\).

The equation we're given is: \((2^r)(4^s) = 16\)

This means we're multiplying 2 raised to the power r by 4 raised to the power s, and this product equals 16. Since r and s must be positive integers (whole numbers greater than zero), we need to find which specific values work.

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

2. Express everything as powers of the same base

To solve this equation easily, let's express everything using the same base. Since 2, 4, and 16 are all powers of 2, let's convert everything to base 2.

First, let's rewrite 4 as a power of 2:
\(4 = 2^2\)

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

Next, let's rewrite 16 as a power of 2:
\(16 = 2^4\)

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

This is much cleaner! Everything is now expressed as powers of 2, which makes the equation much easier to work with.

3. Apply the fundamental exponent rule

When we multiply powers that have the same base, we add the exponents together. This is a fundamental rule of exponents.

So \((2^r)(2^{(2s)}) = 2^{(r + 2s)}\)

Our equation now becomes:
\(2^{(r + 2s)} = 2^4\)

This is beautifully simple! We have the same base (2) on both sides of the equation.

4. Set up the equation for direct comparison

Since both sides of our equation have the same base (2), and the bases are equal, the exponents must also be equal.

This gives us:
\(r + 2s = 4\)

This is now a simple linear equation in two variables, where we need r and s to be positive integers.

5. Solve systematically for positive integer solutions

We need to find positive integer values for r and s that satisfy: \(r + 2s = 4\)

Let's try different positive integer values for s and see what values of r we get:

If \(s = 1\): \(r + 2(1) = 4\), so \(r = 2\)
If \(s = 2\): \(r + 2(2) = 4\), so \(r = 0\)

Since r must be a positive integer, \(r = 0\) doesn't work.
If \(s = 3\) or higher: r would be negative, which doesn't work.

So our only solution is \(r = 2\) and \(s = 1\).

Let's verify: \((2^2)(4^1) = 4 \times 4 = 16\) ✓

Therefore: \(2r + s = 2(2) + 1 = 4 + 1 = 5\)

Process Skill: APPLY CONSTRAINTS - Ensuring our solution satisfies the requirement that r and s are positive integers

Final Answer

\(2r + s = 5\)

The answer is (D) 5.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Students may try to solve by testing answer choices instead of using algebraic manipulation. This approach becomes time-consuming and doesn't build conceptual understanding of exponent rules.

Faltering Point 2: Students may attempt to work with mixed bases (keeping \(4^s\) as \(4^s\) instead of converting to base 2), making the equation much harder to solve and leading to computational errors.

Errors while executing the approach

Faltering Point 1: Students may incorrectly convert \(4^s\) to base 2. A common error is writing \(4^s = 2^s\) instead of the correct \(4^s = (2^2)^s = 2^{(2s)}\).

Faltering Point 2: Students may overlook the constraint that both r and s must be positive integers. They might accept \(r = 0, s = 2\) as a valid solution instead of recognizing that r must be greater than zero.

Faltering Point 3: Students may make arithmetic errors when checking which values work, such as incorrectly calculating that \(2^0 \times 4^2 = 16\), leading them to use invalid values in their final calculation.

Errors while selecting the answer

Faltering Point 1: Students may confuse what the question is asking for and provide the values of r and s individually instead of calculating the expression \(2r + s\). For example, they might select \(r = 2\) or \(s = 1\) instead of computing \(2(2) + 1 = 5\).