Which expression is equivalent to \(\frac{(6\mathrm{x}^2\mathrm{y})(5\mathrm{xy}^3)}{3\mathrm{xy}^2}\)?

1. INFER the solution strategy

• We have a fraction with two monomials multiplied in the numerator and one monomial in the denominator
• Strategy: First multiply the numerators together, then divide by the denominator
• This keeps our work organized and reduces chance of errors

2. SIMPLIFY the numerator by multiplying the monomials

• Multiply \((6\mathrm{x}^2\mathrm{y})(5\mathrm{xy}^3)\):
  - Coefficients: \(6 \times 5 = 30\)
  - x terms: \(\mathrm{x}^2 \times \mathrm{x}^1 = \mathrm{x}^{(2+1)} = \mathrm{x}^3\)
  - y terms: \(\mathrm{y}^1 \times \mathrm{y}^3 = \mathrm{y}^{(1+3)} = \mathrm{y}^4\)
• Numerator becomes: \(30\mathrm{x}^3\mathrm{y}^4\)

3. SIMPLIFY the entire fraction by dividing

• Now we have: \((30\mathrm{x}^3\mathrm{y}^4) \div (3\mathrm{xy}^2)\)
• Divide each part:
  - Coefficients: \(30 \div 3 = 10\)
  - x terms: \(\mathrm{x}^3 \div \mathrm{x}^1 = \mathrm{x}^{(3-1)} = \mathrm{x}^2\)
  - y terms: \(\mathrm{y}^4 \div \mathrm{y}^2 = \mathrm{y}^{(4-2)} = \mathrm{y}^2\)

Answer: \(10\mathrm{x}^2\mathrm{y}^2\) (Choice D)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with exponent rules: Students often confuse when to add versus subtract exponents. They might add exponents when dividing (\(\mathrm{x}^3 \div \mathrm{x} = \mathrm{x}^4\) instead of \(\mathrm{x}^2\)) or subtract when multiplying (\(\mathrm{x}^2 \times \mathrm{x} = \mathrm{x}^1\) instead of \(\mathrm{x}^3\)).

This leads to incorrect variable terms and typically causes confusion since none of the answer choices will match their incorrect result, leading to guessing.

Second Most Common Error:

Poor coefficient calculation during SIMPLIFY: Students might make arithmetic errors with the coefficients, such as calculating \(30 \div 3 = 9\) instead of 10, or forgetting to multiply \(6 \times 5\) in the first step.

This may lead them to select Choice C (\(9\mathrm{x}^2\mathrm{y}^2\)) if they get the exponents right but miscalculate \(30 \div 3 = 9\).

The Bottom Line:

This problem tests systematic application of exponent rules through multiple steps. Students need to stay organized and carefully track both coefficient arithmetic and exponent operations without mixing up the rules for multiplication and division.