{sqrt(x^5)}{sqrt[3]{x^4}} = x^(a/b) If the equation above is true for all positive values of x, what is the value of...

1. TRANSLATE radical expressions to fractional exponents

• Given equation: \(\frac{\sqrt{x^5}}{\sqrt[3]{x^4}} = x^{\frac{a}{b}}\)

• Convert to fractional exponents:
  • \(\sqrt{x^5} = x^{\frac{5}{2}}\) (square root means exponent 1/2)
  • \(\sqrt[3]{x^4} = x^{\frac{4}{3}}\) (cube root means exponent 1/3)

• Rewritten equation: \(\frac{x^{\frac{5}{2}}}{x^{\frac{4}{3}}} = x^{\frac{a}{b}}\)

2. SIMPLIFY using exponent division rules

• Apply the rule: \(\frac{x^m}{x^n} = x^{m-n}\)

• So: \(\frac{x^{\frac{5}{2}}}{x^{\frac{4}{3}}} = x^{\frac{5}{2} - \frac{4}{3}}\)

3. SIMPLIFY the fraction subtraction

• Calculate: \(\frac{5}{2} - \frac{4}{3}\)
• Find common denominator: \(\frac{5}{2} = \frac{15}{6}\) and \(\frac{4}{3} = \frac{8}{6}\)
• Subtract: \(\frac{15}{6} - \frac{8}{6} = \frac{7}{6}\)

4. Match exponents to find a/b

• We have: \(x^{\frac{7}{6}} = x^{\frac{a}{b}}\)
• Therefore: \(\frac{a}{b} = \frac{7}{6}\)

Answer: \(\frac{7}{6}\) (which equals 1.166... or 1.167 when rounded)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not remember how to convert radicals to fractional exponents, particularly getting confused about which root corresponds to which fractional exponent.

For example, they might incorrectly write \(\sqrt[3]{x^4} = x^{\frac{3}{4}}\) instead of \(x^{\frac{4}{3}}\), mixing up the position of the root index and the power. This fundamental conversion error completely derails the solution from the start, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly convert to fractional exponents but make computational errors when subtracting fractions.

A common mistake is incorrectly finding the common denominator or making arithmetic errors: writing \(\frac{5}{2} - \frac{4}{3} = \frac{1}{1} = 1\), or getting \(\frac{11}{6}\) instead of \(\frac{7}{6}\). This leads them to an incorrect final answer that doesn't match any of the given equivalent forms.

The Bottom Line:

This problem tests whether students can fluently move between radical and exponential notation, then apply exponent rules systematically. The key challenge is maintaining accuracy through multiple algebraic steps while working with fractional exponents.