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We need to find the value of \((6^{14} \times 5^{13}) - (6^{13} \times 5^{14})\). Let's think about what this means in plain English: we're subtracting two very large numbers, each made up of powers of 6 and 5 multiplied together. If we tried to calculate these numbers directly, we'd be dealing with astronomical values - \(6^{14}\) alone is over 78 billion! This tells us we need a smarter approach than brute force calculation.
The key insight is recognizing that when we have such large exponential expressions in a difference, there's usually a way to factor and simplify without computing the massive numbers directly.
Process Skill: TRANSLATE - Converting this intimidating expression into a manageable algebraic problem
Let's look at what both terms have in common. In plain English, both expressions contain powers of 6 and powers of 5. The first term has \(6^{14} \times 5^{13}\), and the second has \(6^{13} \times 5^{14}\).
When we want to factor expressions like this, we look for the lowest power of each base that appears in both terms. Here, the lowest power of 6 is \(6^{13}\) (since both terms have at least \(6^{13}\)), and the lowest power of 5 is \(5^{13}\) (since both terms have at least \(5^{13}\)).
This means we can factor out \(6^{13} \times 5^{13}\) from both terms, which will dramatically simplify our calculation.
Now let's actually do the factoring. We can rewrite our original expression by pulling out the common factors:
\((6^{14} \times 5^{13}) - (6^{13} \times 5^{14})\)
Let's factor out \(6^{13} \times 5^{13}\) from both terms:
= \(6^{13} \times 5^{13} \times (6^1) - 6^{13} \times 5^{13} \times (5^1)\)
= \(6^{13} \times 5^{13} \times (6 - 5)\)
= \(6^{13} \times 5^{13} \times 1\)
= \(6^{13} \times 5^{13}\)
Now here's the beautiful part - we can rewrite this as \((6 \times 5)^{13} = 30^{13}\).
Process Skill: SIMPLIFY - Using factoring to avoid calculating enormous numbers
Looking at our result of \(30^{13}\), let's check this against the answer choices:
A. 0 - Not our result
B. 1 - Not our result
C. 30 - This would be \(30^1\), not \(30^{13}\)
D. \(11^{13}\) - Wrong base
E. \(30^{13}\) - This matches exactly!
This confirms our factoring approach was correct. The appearance of \(30^{13}\) as an answer choice also validates that we were meant to recognize that \(6 \times 5 = 30\).
The answer is E. \(30^{13}\)
To verify: \((6^{14} \times 5^{13}) - (6^{13} \times 5^{14}) = 6^{13} \times 5^{13} \times (6 - 5) = 6^{13} \times 5^{13} \times 1 = (6 \times 5)^{13} = 30^{13}\)
1. Attempting brute force calculation instead of factoring: Students often see large exponential expressions like \(6^{14} \times 5^{13}\) and immediately try to calculate these massive numbers directly. This leads to computational nightmares and usually abandoning the problem. The key insight is recognizing that GMAT problems with such large numbers are designed to be solved through algebraic manipulation, not direct calculation.
2. Missing the subtraction structure pattern: When students see \((6^{14} \times 5^{13}) - (6^{13} \times 5^{14})\), they may not immediately recognize this as a classic 'difference of products' that can be factored. Instead, they might look for other patterns like difference of squares or try to manipulate each term separately, missing the opportunity to factor out common terms.
3. Not identifying the common factor systematically: Students may recognize they need to factor but fail to systematically identify the highest common factor. They might factor out \(6^{13}\) but miss \(5^{13}\), or vice versa, leading to incomplete factoring and incorrect simplification.
1. Incorrect exponent manipulation when factoring: When factoring out \(6^{13} \times 5^{13}\), students often make errors with exponent rules. For example, they might write \(6^{14} = 6^{13} \times 6^1\) correctly but then incorrectly handle \(5^{14} = 5^{13} \times 5^1\), or they might subtract exponents instead of factoring properly.
2. Arithmetic error in the final subtraction: Even after correctly factoring to get \(6^{13} \times 5^{13} \times (6 - 5)\), students sometimes make the simple arithmetic error of calculating 6 - 5 incorrectly, especially when working under time pressure with such complex-looking expressions.
3. Forgetting to apply the power rule: Students may correctly arrive at \(6^{13} \times 5^{13} \times 1 = 6^{13} \times 5^{13}\) but then fail to recognize that this can be written as \((6 \times 5)^{13}\). They might leave their answer as \(6^{13} \times 5^{13}\) and not see how this connects to the answer choices.
1. Confusing \(30^{13}\) with 30: After correctly deriving \(30^{13}\), students might see answer choice C (30) and select it, forgetting about the exponent 13. This is especially likely if they're rushing or if they calculated \(6 \times 5 = 30\) as their final step without keeping track of the power.
2. Second-guessing the large exponent: Students might correctly calculate \(30^{13}\) but then doubt themselves because it seems like such a large number. They may think there must be further simplification possible and incorrectly select a 'simpler' answer like choice A (0) or choice B (1).