Which expression is equivalent to 5psqrt(p) + 15p/(sqrt(p)), where p is a positive number?20p20psqrt(p)\(5(\mathrm{p}+3)\sqrt{\mathrm{p}}\)\(\frac{5\m...

1. TRANSLATE the problem information

• Given expression: \(5p\sqrt{p} + \frac{15p}{\sqrt{p}}\)
• Need to find equivalent expression from the choices

2. INFER the approach needed

• The second term \(\frac{15p}{\sqrt{p}}\) has a radical in the denominator - this suggests we should convert it to have radicals in the numerator instead
• Once both terms have similar radical forms, we can likely factor them

3. SIMPLIFY the second term using exponent rules

• Convert to exponential form: \(\frac{15p}{\sqrt{p}} = \frac{15p^1}{p^{1/2}}\)
• Apply exponent rule \(\frac{a^m}{a^n} = a^{m-n}\): \(15p^{1-1/2} = 15p^{1/2}\)
• Convert back to radical form: \(15p^{1/2} = 15\sqrt{p}\)

4. SIMPLIFY by substituting back

• Original expression becomes: \(5p\sqrt{p} + 15\sqrt{p}\)
• Now both terms contain \(\sqrt{p}\), making factoring possible

5. INFER the factoring strategy and SIMPLIFY

• Both terms have \(5\sqrt{p}\) as a common factor
• \(5p\sqrt{p} = 5\sqrt{p} \cdot p\) and \(15\sqrt{p} = 5\sqrt{p} \cdot 3\)
• Factor out \(5\sqrt{p}\): \(5\sqrt{p}(p + 3)\)
• Rewrite in standard form: \(5(p + 3)\sqrt{p}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Incorrectly handling the division \(\frac{15p}{\sqrt{p}}\)

Students often struggle with dividing by radicals and may make errors like:

• Thinking \(\frac{15p}{\sqrt{p}} = \frac{15p}{p} = 15\) (ignoring the radical)
• Or trying \(\frac{15p}{\sqrt{p}} = \frac{15\sqrt{p}}{p}\) (incorrect radical manipulation)

This leads to wrong intermediate expressions that don't match any answer choice, causing confusion and guessing.

Second Most Common Error:

Poor INFER reasoning about factoring: After correctly getting \(5p\sqrt{p} + 15\sqrt{p}\), not recognizing the common factor

Students see the two different-looking terms \(5p\sqrt{p}\) and \(15\sqrt{p}\) and may think they can't be combined, or attempt to add them incorrectly (like \(5p\sqrt{p} + 15\sqrt{p} = 20p\sqrt{p}\)).

This may lead them to select Choice B (\(20p\sqrt{p}\)).

The Bottom Line:

This problem tests whether students can confidently manipulate expressions with radicals using exponent rules, then recognize factoring opportunities. The key insight is that dividing by a radical creates an opportunity to rewrite expressions in a form that reveals common factors.