- Translate the problem requirements: We have an equation with two exponential terms that differ by 3 in their exponents, and we need to find the value of the variable x that makes this equation equal 2,106
- Factor out the common exponential term: Recognize that both terms share a common factor and simplify the left side of the equation
- Isolate the remaining exponential expression: Solve for the simpler exponential term by dividing both sides appropriately
- Convert to recognizable power of 3: Identify which power of 3 equals our isolated value and solve for x
Execution of Strategic Approach
1. Translate the problem requirements
Let's start by understanding what we have. We're given an equation with two exponential terms: \(\mathrm{3^x}\) and \(\mathrm{3^{(x-3)}}\). The key thing to notice is that these exponents differ by exactly 3.
Think of it this way: if x = 10, then we'd have \(\mathrm{3^{10}}\) and \(\mathrm{3^7}\). The second term has an exponent that's 3 less than the first term's exponent.
Our equation is: \(\mathrm{3^x - 3^{(x-3)} = 2{,}106}\)
We need to find the value of x that makes this equation true.
Process Skill: TRANSLATE - Converting the exponential relationship into clear mathematical understanding
2. Factor out the common exponential term
Here's where we use a clever insight. Both terms share a common factor. Let me explain this in everyday terms first.
Think about \(\mathrm{3^x}\) and \(\mathrm{3^{(x-3)}}\). The second term can be written as \(\mathrm{3^x ÷ 3^3}\), because when we divide powers with the same base, we subtract the exponents: \(\mathrm{3^x ÷ 3^3 = 3^{(x-3)}}\).
Since \(\mathrm{3^3 = 27}\), we can rewrite \(\mathrm{3^{(x-3)}}\) as \(\mathrm{3^x ÷ 27}\).
So our equation becomes:
\(\mathrm{3^x - (3^x ÷ 27) = 2{,}106}\)
Now we can factor out \(\mathrm{3^x}\) from both terms:
\(\mathrm{3^x(1 - \frac{1}{27}) = 2{,}106}\)
Simplifying the fraction: \(\mathrm{1 - \frac{1}{27} = \frac{26}{27}}\)
So we have: \(\mathrm{3^x × \frac{26}{27} = 2{,}106}\)
3. Isolate the remaining exponential expression
Now we need to solve for \(\mathrm{3^x}\) by getting it by itself.
We have: \(\mathrm{3^x × \frac{26}{27} = 2{,}106}\)
To isolate \(\mathrm{3^x}\), we divide both sides by \(\mathrm{\frac{26}{27}}\). Remember, dividing by a fraction is the same as multiplying by its reciprocal:
\(\mathrm{3^x = 2{,}106 ÷ \frac{26}{27}}\)
\(\mathrm{3^x = 2{,}106 × \frac{27}{26}}\)
Let's calculate this step by step:
\(\mathrm{3^x = 2{,}106 × 27 ÷ 26}\)
\(\mathrm{3^x = 56{,}862 ÷ 26}\)
\(\mathrm{3^x = 2{,}187}\)
4. Convert to recognizable power of 3
Now we need to figure out what power of 3 equals 2,187.
Let's think about powers of 3:
\(\mathrm{3^1 = 3}\)
\(\mathrm{3^2 = 9}\)
\(\mathrm{3^3 = 27}\)
\(\mathrm{3^4 = 81}\)
\(\mathrm{3^5 = 243}\)
\(\mathrm{3^6 = 729}\)
\(\mathrm{3^7 = 2{,}187}\)
Perfect! We found that \(\mathrm{3^7 = 2{,}187}\).
Therefore, x = 7.
Process Skill: INFER - Recognizing that we need to work backwards from 2,187 to find the corresponding power of 3
Final Answer
The value of x is 7.
Let's verify: If x = 7, then:
\(\mathrm{3^7 - 3^{(7-3)} = 3^7 - 3^4 = 2{,}187 - 81 = 2{,}106}\) ✓
The answer is D. 7
Common Faltering Points
Errors while devising the approach
- Missing the common factor insight: Students often don't recognize that \(\mathrm{3^{(x-3)}}\) can be rewritten as \(\mathrm{3^x ÷ 3^3}\). Instead, they might try to solve the equation by testing answer choices immediately or attempting to isolate each term separately, making the problem much more difficult than necessary.
- Confusion with exponent rules: Students may incorrectly think that \(\mathrm{3^x - 3^{(x-3)}}\) simplifies to \(\mathrm{3^{(x-(x-3))} = 3^3}\), not understanding that subtraction of exponential terms doesn't follow the same rules as multiplication or division of exponential terms.
- Overlooking the factoring opportunity: Students might not see that both terms share \(\mathrm{3^{(x-3)}}\) as a common factor, missing the chance to factor out this term and simplify the equation to \(\mathrm{3^{(x-3)}(3^3 - 1) = 2{,}106}\).
Errors while executing the approach
- Arithmetic errors in fraction calculation: When computing \(\mathrm{1 - \frac{1}{27}}\), students often make mistakes getting \(\mathrm{\frac{25}{27}}\) instead of \(\mathrm{\frac{26}{27}}\), or when multiplying \(\mathrm{2{,}106 × \frac{27}{26}}\), they might incorrectly calculate \(\mathrm{56{,}862 ÷ 26 = 2{,}187}\).
- Incorrect reciprocal handling: When dividing by the fraction \(\mathrm{\frac{26}{27}}\), students may forget to flip the fraction and instead multiply by \(\mathrm{\frac{26}{27}}\) rather than \(\mathrm{\frac{27}{26}}\), leading to an incorrect value for \(\mathrm{3^x}\).
- Powers of 3 calculation errors: Students might make computational mistakes when calculating successive powers of 3, particularly for larger values like \(\mathrm{3^6 = 729}\) or \(\mathrm{3^7 = 2{,}187}\), leading them to the wrong value of x.
Errors while selecting the answer
- Verification step omission: Students might arrive at x = 7 but fail to substitute back into the original equation to verify their answer, missing the chance to catch any computational errors made during the solution process.
- Misreading the final result: After calculating that \(\mathrm{3^x = 2{,}187}\), students might confuse this result with the final answer, potentially selecting a choice related to 2,187 instead of recognizing that x = 7.
