Which expression is equivalent to 18x^2/3^x + 18x^2 * 3^x, where x gt 0?
GMAT Advanced Math : (Adv_Math) Questions
Which expression is equivalent to \(\frac{18\mathrm{x}^2}{3^\mathrm{x}} + 18\mathrm{x}^2 \cdot 3^\mathrm{x}\), where \(\mathrm{x} \gt 0\)?
\(\frac{36\mathrm{x}^2}{3^\mathrm{x}}\)
\(\frac{18\mathrm{x}^2(3^{2\mathrm{x}})}{3^\mathrm{x}}\)
\(\frac{18\mathrm{x}^2(1 + 3^{2\mathrm{x}})}{3^\mathrm{x}}\)
\(36\mathrm{x}^2 \cdot 3^\mathrm{x}\)
\(\frac{18\mathrm{x}^2(1 + 3^\mathrm{x})}{3^\mathrm{x}}\)
1. TRANSLATE the problem information
- Given expression: \(\frac{18x^2}{3^x} + 18x^2 \cdot 3^x\), where \(x \gt 0\)
- Goal: Find an equivalent expression from the answer choices
2. INFER the approach needed
- To combine these terms, I need a common denominator
- The first term already has \(3^x\) in the denominator
- I need to rewrite the second term to have the same denominator
3. SIMPLIFY by creating equivalent fractions
- Rewrite the second term with denominator \(3^x\):
\(18x^2 \cdot 3^x = 18x^2 \cdot 3^x \cdot \frac{3^x}{3^x}\)
\(= \frac{18x^2 \cdot 3^{2x}}{3^x}\)
- Note: \(3^x \cdot 3^x = 3^{x+x} = 3^{2x}\) using exponent rules
4. SIMPLIFY by combining the fractions
- Now both terms have the same denominator:
\(\frac{18x^2}{3^x} + \frac{18x^2 \cdot 3^{2x}}{3^x}\)
\(= \frac{18x^2 + 18x^2 \cdot 3^{2x}}{3^x}\)
5. SIMPLIFY by factoring
- Factor out the common factor \(18x^2\):
\(\frac{18x^2 + 18x^2 \cdot 3^{2x}}{3^x}\)
\(= \frac{18x^2(1 + 3^{2x})}{3^x}\)
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize the need for a common denominator and try to combine the terms directly, treating addition of fractions like multiplication.
They might think: \(\frac{18x^2}{3^x} + 18x^2 \cdot 3^x = 18x^2(\text{something with } 3^x)\), leading to choices like (A) or (D) that don't properly account for both terms.
This may lead them to select Choice A (\(\frac{36x^2}{3^x}\)) or Choice D (\(36x^2 \cdot 3^x\)).
Second Most Common Error:
Poor SIMPLIFY execution: Students make errors with exponent rules when creating the common denominator, particularly getting confused about \(3^x \cdot 3^x = 3^{2x}\).
They might incorrectly compute \(3^x \cdot 3^x = 3^x\) or \(3^x \cdot 3^x = 3^{x^2}\), leading to wrong expressions that don't match the answer choices exactly.
This leads to confusion and guessing among the remaining choices.
The Bottom Line:
This problem requires both strategic thinking about how to combine different types of terms (rational expression + polynomial) and careful execution of exponent rules - many students struggle with one or both of these elements.
\(\frac{36\mathrm{x}^2}{3^\mathrm{x}}\)
\(\frac{18\mathrm{x}^2(3^{2\mathrm{x}})}{3^\mathrm{x}}\)
\(\frac{18\mathrm{x}^2(1 + 3^{2\mathrm{x}})}{3^\mathrm{x}}\)
\(36\mathrm{x}^2 \cdot 3^\mathrm{x}\)
\(\frac{18\mathrm{x}^2(1 + 3^\mathrm{x})}{3^\mathrm{x}}\)