The expression (10sqrt(50x^(15)))/(sqrt(2x^(4))) is equivalent to the expression ax^(b), where a and b are constants and x gt 1. What...

1. TRANSLATE the problem information

• Given: \(\frac{10\sqrt{50\mathrm{x}^{15}}}{\sqrt{2\mathrm{x}^4}}\) must be written as \(\mathrm{ax}^\mathrm{b}\)
• Find: The value of \(\mathrm{a - b}\)

2. INFER the approach

• The key insight is to combine the two separate radicals into one expression first
• Then simplify what's inside the radical before separating constants from variables

3. SIMPLIFY by combining radicals

Use the property \(\frac{\sqrt{\mathrm{m}}}{\sqrt{\mathrm{n}}} = \sqrt{\frac{\mathrm{m}}{\mathrm{n}}}\):

\(\frac{10\sqrt{50\mathrm{x}^{15}}}{\sqrt{2\mathrm{x}^4}} = 10\sqrt{\frac{50\mathrm{x}^{15}}{2\mathrm{x}^4}}\)

4. SIMPLIFY the fraction inside the radical

• Coefficient: \(50 \div 2 = 25\)
• Variable: \(\mathrm{x}^{15} \div \mathrm{x}^4 = \mathrm{x}^{(15-4)} = \mathrm{x}^{11}\)

Result: \(10\sqrt{25\mathrm{x}^{11}}\)

5. SIMPLIFY by separating the radical

Use \(\sqrt{\mathrm{mn}} = \sqrt{\mathrm{m}} \times \sqrt{\mathrm{n}}\):

\(10\sqrt{25\mathrm{x}^{11}} = 10\sqrt{25} \times \sqrt{\mathrm{x}^{11}} = 10 \times 5 \times \sqrt{\mathrm{x}^{11}} = 50\sqrt{\mathrm{x}^{11}}\)

6. INFER the need to convert to fractional exponent

Since we need the form \(\mathrm{ax}^\mathrm{b}\), convert \(\sqrt{\mathrm{x}^{11}} = \mathrm{x}^{\frac{11}{2}}\)

Final simplified form: \(50\mathrm{x}^{\frac{11}{2}}\)

7. TRANSLATE to identify constants

Comparing \(50\mathrm{x}^{\frac{11}{2}}\) to \(\mathrm{ax}^\mathrm{b}\):

• \(\mathrm{a} = 50\)
• \(\mathrm{b} = \frac{11}{2}\)

8. SIMPLIFY to find the final answer

\(\mathrm{a - b} = 50 - \frac{11}{2}\)

\(= \frac{100}{2} - \frac{11}{2}\)

\(= \frac{89}{2}\)

Answer: \(\frac{89}{2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when simplifying the fraction \(\frac{50\mathrm{x}^{15}}{2\mathrm{x}^4}\), either miscalculating \(50\div2 = 25\) or incorrectly applying the exponent rule to get \(\mathrm{x}^7\) instead of \(\mathrm{x}^{11}\).

This leads to an incorrect simplified expression, which then cascades through the remaining steps. For instance, getting \(10\sqrt{20\mathrm{x}^7}\) instead of \(10\sqrt{25\mathrm{x}^{11}}\) would eventually lead to \(\mathrm{a} = 20\sqrt{5}\) and a completely different final answer.

Second Most Common Error:

Poor INFER reasoning about radical manipulation: Students attempt to simplify each radical separately instead of combining them first, leading to unnecessarily complex intermediate steps like trying to simplify \(10\sqrt{50\mathrm{x}^{15}}\) individually.

This approach makes the problem much more difficult and often leads to confusion about how to proceed, causing students to abandon systematic solution and guess.

The Bottom Line:

This problem tests whether students can systematically apply multiple radical properties in the correct sequence while maintaining accuracy through several algebraic steps.