- Translate the problem requirements: We need to find nonnegative integers x and y such that when we factor the left side expression \(2,008^{2,008} - 2,008^{2,007}\), it equals \((2,008^{\mathrm{x}})(2,007^{\mathrm{y}})\). Then we calculate \(\mathrm{x} + \mathrm{y}\).
- Factor the left side by identifying the common term: Look for the greatest common factor in the expression \(2,008^{2,008} - 2,008^{2,007}\) to simplify it into a product form.
- Rewrite the factored form to match the required format: Express the factored result in the form \((2,008^{\mathrm{x}})(2,007^{\mathrm{y}})\) to identify the values of x and y.
- Calculate the final answer: Add the values of x and y to get the required sum.
Execution of Strategic Approach
1. Translate the problem requirements
Let's understand what we're looking for. We have an equation that says:
\(2,008^{2,008} - 2,008^{2,007} = (2,008^{\mathrm{x}})(2,007^{\mathrm{y}})\)
We need to find the values of x and y (both nonnegative integers), and then calculate \(\mathrm{x} + \mathrm{y}\).
The key insight is that the left side of the equation needs to be rewritten in a form that looks like the right side - a product involving powers of 2,008 and 2,007.
Process Skill: TRANSLATE - Converting the algebraic equation into a clear goal of finding specific integer values
2. Factor the left side by identifying the common term
Let's look at the left side: \(2,008^{2,008} - 2,008^{2,007}\)
Both terms have a common factor. Think of this like factoring out a common term from two numbers. For example, if we had \(12 - 6\), we could factor out 6 to get \(6(2 - 1) = 6(1)\).
Here, both terms contain \(2,008^{2,007}\). We can factor this out:
- \(2,008^{2,008} = 2,008^{2,007} \times 2,008^1 = 2,008^{2,007} \times 2,008\)
- \(2,008^{2,007} = 2,008^{2,007} \times 1\)
So: \(2,008^{2,008} - 2,008^{2,007} = 2,008^{2,007} \times 2,008 - 2,008^{2,007} \times 1\)
Factoring out \(2,008^{2,007}\):
\(2,008^{2,008} - 2,008^{2,007} = 2,008^{2,007}(2,008 - 1) = 2,008^{2,007} \times 2,007\)
Process Skill: SIMPLIFY - Recognizing and extracting common factors to reduce complexity
3. Rewrite the factored form to match the required format
Now we have: \(2,008^{2,007} \times 2,007\)
We need this to match the form \((2,008^{\mathrm{x}})(2,007^{\mathrm{y}})\).
Let's rewrite our result:
\(2,008^{2,007} \times 2,007 = 2,008^{2,007} \times 2,007^1\)
Comparing this with \((2,008^{\mathrm{x}})(2,007^{\mathrm{y}})\):
- The power of 2,008 is 2,007, so \(\mathrm{x} = 2,007\)
- The power of 2,007 is 1, so \(\mathrm{y} = 1\)
Let's verify: \((2,008^{2,007})(2,007^1) = 2,008^{2,007} \times 2,007\) ✓
4. Calculate the final answer
We found that \(\mathrm{x} = 2,007\) and \(\mathrm{y} = 1\).
Therefore: \(\mathrm{x} + \mathrm{y} = 2,007 + 1 = 2,008\)
Looking at our answer choices:
- 1
- 2
- 2,007
- 2,008
- 4,015
Our answer is 2,008, which corresponds to choice (D).
Common Faltering Points
Errors while devising the approach
- Misinterpreting the equation structure: Students may try to expand or manipulate both sides of the equation simultaneously instead of recognizing that they need to factor the left side first. They might attempt to distribute the exponents or apply logarithms, missing the simpler factoring approach.
- Not recognizing the factoring opportunity: Students may fail to see that \(2,008^{2,008}\) and \(2,008^{2,007}\) share a common factor. They might try complex algebraic manipulations when the solution requires basic factoring of exponential expressions.
- Overlooking the constraint significance: Students may not pay sufficient attention to the constraint that x and y must be nonnegative integers, potentially leading them to consider approaches that could yield fractional or negative solutions.
Errors while executing the approach
- Incorrect factoring execution: When factoring out \(2,008^{2,007}\), students may incorrectly write the remaining terms. For example, they might write \((2,008 + 1)\) instead of \((2,008 - 1)\), or make errors in the exponent arithmetic when separating \(2,008^{2,008}\) into \(2,008^{2,007} \times 2,008\).
- Arithmetic error in subtraction: Students may incorrectly calculate \(2,008 - 1\), potentially writing 2,009 instead of 2,007, which would lead to an entirely wrong final answer.
Errors while selecting the answer
- Providing individual values instead of the sum: Students may correctly find \(\mathrm{x} = 2,007\) and \(\mathrm{y} = 1\) but then select answer choice (C) 2,007 (the value of x) instead of calculating \(\mathrm{x} + \mathrm{y} = 2,008\). This happens when students lose track of what the question is actually asking for.
- Mixing up the variable assignments: Students might correctly identify the powers but assign them to the wrong variables \((\mathrm{x} = 1, \mathrm{y} = 2,007)\), leading to the same sum but potentially causing confusion if they double-check their work incorrectly.
