Which expression represents the product of 2a^(-3)b^(4)a^(5) and a^(6)b^(-2) - b^(7)a^(-8), written using only positive exponents?2a^(8)b^(2) - 2a^(6)...
GMAT Advanced Math : (Adv_Math) Questions
- \(2\mathrm{a}^{8}\mathrm{b}^{2} - 2\mathrm{a}^{6}\mathrm{b}^{11}\)
- \(2\mathrm{a}^{8}\mathrm{b}^{2} - \frac{2\mathrm{b}^{11}}{\mathrm{a}^{6}}\)
- \(2\mathrm{a}^{8}\mathrm{b}^{2} + \frac{2\mathrm{b}^{11}}{\mathrm{a}^{6}}\)
- \(\mathrm{a}^{8}\mathrm{b}^{2} - \frac{\mathrm{b}^{11}}{\mathrm{a}^{6}}\)
- \(2\mathrm{a}^{8}\mathrm{b}^{2} - 2\mathrm{a}^{-6}\mathrm{b}^{11}\)
1. SIMPLIFY the first factor
- Given: \(\mathrm{(2a^{-3}b^4a^5)}\) and \(\mathrm{(a^6b^{-2} - b^7a^{-8})}\)
- First, combine like bases in the first expression: \(\mathrm{a^{-3} \cdot a^5 = a^{-3+5} = a^2}\)
- This gives us: \(\mathrm{2a^2b^4}\)
2. INFER the multiplication strategy
- We need to multiply \(\mathrm{2a^2b^4}\) by the entire binomial \(\mathrm{(a^6b^{-2} - b^7a^{-8})}\)
- This requires distributing: multiply the first factor by each term in the second factor
3. SIMPLIFY using the distributive property
- Set up: \(\mathrm{(2a^2b^4)(a^6b^{-2}) - (2a^2b^4)(b^7a^{-8})}\)
4. SIMPLIFY each product separately
- First product: \(\mathrm{(2a^2b^4)(a^6b^{-2})}\)
- Multiply coefficients: 2
- Add exponents for a: \(\mathrm{2 + 6 = 8}\)
- Add exponents for b: \(\mathrm{4 + (-2) = 2}\)
- Result: \(\mathrm{2a^8b^2}\)
- Second product: \(\mathrm{(2a^2b^4)(b^7a^{-8})}\)
- Multiply coefficients: 2
- Add exponents for a: \(\mathrm{2 + (-8) = -6}\)
- Add exponents for b: \(\mathrm{4 + 7 = 11}\)
- Result: \(\mathrm{2a^{-6}b^{11}}\)
5. SIMPLIFY to convert negative exponents
- Current expression: \(\mathrm{2a^8b^2 - 2a^{-6}b^{11}}\)
- Convert \(\mathrm{a^{-6}}\) to positive: \(\mathrm{a^{-6} = \frac{1}{a^6}}\)
- Final form: \(\mathrm{2a^8b^2 - \frac{2b^{11}}{a^6}}\)
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students incorrectly add or subtract exponents instead of following the proper rules
Many students might calculate the first factor as \(\mathrm{2a^{-3+4+5} = 2a^6b^4}\), forgetting that only like bases get their exponents added. Or they might subtract exponents when multiplying: \(\mathrm{a^2 \cdot a^6 = a^{2-6} = a^{-4}}\). These errors compound through the problem.
This may lead them to select Choice A (\(\mathrm{2a^8b^2 - 2a^6b^{11}}\)) or get confused and guess.
Second Most Common Error:
Inadequate SIMPLIFY execution with negative exponents: Students correctly work through most steps but fail to properly convert negative exponents to positive form
They might leave their final answer as \(\mathrm{2a^8b^2 - 2a^{-6}b^{11}}\) without converting to positive exponents, or incorrectly convert \(\mathrm{a^{-6}}\) as \(\mathrm{-a^6}\) instead of \(\mathrm{\frac{1}{a^6}}\).
This may lead them to select Choice E (\(\mathrm{2a^8b^2 - 2a^{-6}b^{11}}\)).
The Bottom Line:
This problem tests sustained accuracy with exponent rules across multiple algebraic steps. Success requires careful attention to both the laws of exponents and the requirement to express the final answer using only positive exponents.