Which expression is equivalent to p^(14) r^9/p^5 r^(20), where p gt 0 and r gt 0?

1. INFER the approach needed

• We have a fraction with the same bases (p and r) in numerator and denominator
• This signals we should use the quotient rule for exponents
• We'll handle each variable separately

2. SIMPLIFY using the quotient rule for the p terms

• Apply: \(\frac{\mathrm{p}^{14}}{\mathrm{p}^5} = \mathrm{p}^{14-5} = \mathrm{p}^9\)
• The p part of our answer is \(\mathrm{p}^9\)

3. SIMPLIFY using the quotient rule for the r terms

• Apply: \(\frac{\mathrm{r}^9}{\mathrm{r}^{20}} = \mathrm{r}^{9-20} = \mathrm{r}^{-11}\)
• We get a negative exponent, which needs conversion

4. SIMPLIFY the negative exponent

• Convert: \(\mathrm{r}^{-11} = \frac{1}{\mathrm{r}^{11}}\)
• The r part becomes \(\frac{1}{\mathrm{r}^{11}}\)

5. SIMPLIFY by combining the results

• We have: \(\mathrm{p}^9 \cdot \frac{1}{\mathrm{r}^{11}} = \frac{\mathrm{p}^9}{\mathrm{r}^{11}}\)

Answer: A. \(\frac{\mathrm{p}^9}{\mathrm{r}^{11}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly start with the quotient rule but make arithmetic errors when subtracting exponents, especially with the r terms where 9-20 = -11. Some students might calculate this as -9 or +11, leading to incorrect expressions.

This may lead them to select Choice C (\(\frac{\mathrm{p}^7}{\mathrm{r}^9}\)) if they made multiple arithmetic errors.

Second Most Common Error:

Missing conceptual knowledge about negative exponents: Students correctly calculate \(\mathrm{r}^{-11}\) but don't know how to convert it to \(\frac{1}{\mathrm{r}^{11}}\). They might leave it as \(\mathrm{p}^9 \mathrm{r}^{-11}\) or incorrectly think the negative means the entire expression is negative.

This may lead them to select Choice B (\(\mathrm{p}^9 \mathrm{r}^{11}\)) if they drop the negative sign entirely.

The Bottom Line:

Success requires careful arithmetic with signed numbers and solid understanding of negative exponent conversion. The most reliable approach is working slowly through each variable separately and double-checking the subtraction steps.