Which expression is equivalent to the expression below, assuming a neq 0 and b neq 0?63a^9b^7/7a^3b^29a^4b^39a^6b^556a^6b^59a^(12)b^9

1. TRANSLATE the problem structure

• Given: \(\frac{63\mathrm{a}^9\mathrm{b}^7}{7\mathrm{a}^3\mathrm{b}^2}\)
• What this tells us: We have a fraction where both numerator and denominator contain coefficients and variables with exponents

2. SIMPLIFY each component separately

• Handle coefficients first: \(63 \div 7 = 9\)
• Apply quotient rule to 'a' terms: \(\mathrm{a}^9 \div \mathrm{a}^3 = \mathrm{a}^{(9-3)} = \mathrm{a}^6\)
• Apply quotient rule to 'b' terms: \(\mathrm{b}^7 \div \mathrm{b}^2 = \mathrm{b}^{(7-2)} = \mathrm{b}^5\)

3. SIMPLIFY by combining results

• Putting it all together: \(9\mathrm{a}^6\mathrm{b}^5\)

Answer: B (\(9\mathrm{a}^6\mathrm{b}^5\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students confuse multiplication and division rules for exponents, adding exponents instead of subtracting them.

Instead of \(\mathrm{a}^9 \div \mathrm{a}^3 = \mathrm{a}^{(9-3)} = \mathrm{a}^6\), they calculate \(\mathrm{a}^9 \div \mathrm{a}^3 = \mathrm{a}^{(9+3)} = \mathrm{a}^{12}\), and similarly \(\mathrm{b}^7 \div \mathrm{b}^2 = \mathrm{b}^{(7+2)} = \mathrm{b}^9\).

This leads them to select Choice D (\(9\mathrm{a}^{12}\mathrm{b}^9\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students make basic arithmetic errors when dividing the coefficients.

They might calculate \(63 \div 7 = 8\) instead of 9, then correctly apply exponent rules to get \(8\mathrm{a}^6\mathrm{b}^5\). Since this exact answer isn't available, they get confused and may guess or select the closest-looking option.

The Bottom Line:

This problem tests whether students can systematically break down a rational expression and correctly apply the quotient rule for exponents. Success depends on careful execution of basic rules rather than complex conceptual insights.