3^(41) + 3^(42) + 3^(43) =

1. Translate the problem requirements: We need to find the sum \(3^{41} + 3^{42} + 3^{43}\) and express it in the form given by one of the answer choices, which all involve factoring out a power of 3.
2. Identify the common factor strategy: Since all terms are powers of 3, we can factor out the smallest power (\(3^{41}\)) from each term to simplify the expression.
3. Factor out the common term: Extract \(3^{41}\) from each term and simplify what remains inside the parentheses.
4. Calculate the simplified coefficient: Evaluate the remaining terms to get the final coefficient that multiplies \(3^{41}\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to do. We have three numbers that are powers of 3: \(3^{41}\), \(3^{42}\), and \(3^{43}\). We need to add them together and then see which of the given answer choices matches our result.

Looking at the answer choices, they all have something in common - they're written as a number multiplied by either \(3^{41}\) or \(3^{42}\). This tells us that we're probably going to need to factor our sum, which means taking out a common piece from all three terms.

Think of it like this: if you had \(2 \times 5 + 3 \times 5 + 4 \times 5\), you could factor out the 5 to get \(5 \times (2 + 3 + 4) = 5 \times 9\). We're going to do something similar with powers of 3.

Process Skill: TRANSLATE

2. Identify the common factor strategy

Now let's think about what all three terms have in common. We have \(3^{41}\), \(3^{42}\), and \(3^{43}\).

The key insight is that \(3^{42} = 3^{41} \times 3^1\), and \(3^{43} = 3^{41} \times 3^2\). In everyday terms, if \(3^{41}\) is our "basic unit," then \(3^{42}\) is "3 times that unit" and \(3^{43}\) is "9 times that unit" (since \(3^2 = 9\)).

So we can think of our sum as:

• One unit of \(3^{41}\)
• Plus 3 units of \(3^{41}\)
• Plus 9 units of \(3^{41}\)

This means we can factor out \(3^{41}\) from all three terms, just like factoring out a common number from regular arithmetic.

3. Factor out the common term

Let's actually do the factoring now. We want to take \(3^{41}\) out of each term:

From \(3^{41}\): We get \(3^{41} \times 1\)
From \(3^{42}\): We get \(3^{41} \times 3^1 = 3^{41} \times 3\)
From \(3^{43}\): We get \(3^{41} \times 3^2 = 3^{41} \times 9\)

So our original sum becomes:
\(3^{41} + 3^{42} + 3^{43} = 3^{41} \times 1 + 3^{41} \times 3 + 3^{41} \times 9\)

Now we can factor out the \(3^{41}\):
\(= 3^{41} \times (1 + 3 + 9)\)

4. Calculate the simplified coefficient

Now we just need to add up what's inside the parentheses:
\(1 + 3 + 9 = 13\)

So our final answer is:
\(3^{41} + 3^{42} + 3^{43} = 13 \times 3^{41} = 13(3^{41})\)

Looking at our answer choices, this matches choice B exactly!

Final Answer

The answer is B. \(13(3^{41})\)

We can verify this makes sense: we factored out the smallest power of 3 (which was \(3^{41}\)) and found that the remaining coefficient was 13. This matches choice B perfectly, and we can see that the other choices would give us different coefficients (like 7, 3, or 5) which don't match our calculation of \(1 + 3 + 9 = 13\).

Common Faltering Points

Errors while devising the approach

• Missing the factoring strategy: Students may attempt to calculate each power of 3 individually (\(3^{41}\), \(3^{42}\), \(3^{43}\)) rather than recognizing that factoring out the common term is the key approach. This leads to unnecessarily complex calculations with very large numbers.
• Choosing the wrong common factor: Students might try to factor out \(3^{42}\) or \(3^{43}\) instead of the smallest power (\(3^{41}\)). While this can work mathematically, it makes the problem more complex and doesn't match the answer format given in the choices.

Errors while executing the approach

• Incorrect exponent rules application: Students may struggle with rewriting \(3^{42}\) as \(3^{41} \times 3^1\) and \(3^{43}\) as \(3^{41} \times 3^2\). They might incorrectly write these as \(3^{41} \times 3^{41}\) or make other exponent errors.
• Arithmetic errors in coefficient calculation: When adding \(1 + 3 + 9\), students might miscalculate and get 12 or 14 instead of 13, leading them to select an incorrect answer choice.

Errors while selecting the answer

• Confusing the factored form: Even after correctly calculating \(13(3^{41})\), students might mistakenly select answer choice E: \(7(3^{42})\) thinking they need to express their answer in terms of the middle power, or they might not carefully check which power of 3 their coefficient should be multiplied by.