To simplify the given expression, follow these steps:

Expression:

$\frac{\sqrt[9]{a^7b^6}}{\sqrt{ab}}$

Step 1: Write the radicals as exponents

• Recall that $\sqrt[n]{x} = x^{1/n}$.
• Rewrite the expression:

$\frac{(a^7b^6)^{1/9}}{(ab)^{1/2}}$

Step 2: Distribute the exponents

• Apply the exponent $\frac{1}{9}$ to $a^7$ and $b^6$ in the numerator:

$\frac{a^{7/9}b^{6/9}}{a^{1/2}b^{1/2}}$

• Simplify $6/9$ in the numerator:

$\frac{a^{7/9}b^{2/3}}{a^{1/2}b^{1/2}}$

Step 3: Simplify the exponents

• Use the rule $x^m / x^n = x^{m-n}$ to simplify the powers of $a$ and $b$:

$a^{7/9 - 1/2}b^{2/3 - 1/2}$

Step 4: Find common denominators

• For $7/9 - 1/2$, find the common denominator (18):

$\frac{7}{9} = \frac{14}{18}, \quad \frac{1}{2} = \frac{9}{18}, \quad \text{so } 7/9 - 1/2 = \frac{14}{18} - \frac{9}{18} = \frac{5}{18}.$

• For $2/3 - 1/2$, find the common denominator (6):

$\frac{2}{3} = \frac{4}{6}, \quad \frac{1}{2} = \frac{3}{6}, \quad \text{so } 2/3 - 1/2 = \frac{4}{6} - \frac{3}{6} = \frac{1}{6}.$

Step 5: Combine the results

• The simplified expression becomes:

$a^{5/18}b^{1/6}.$

Step 6: Convert back to radical notation

• Rewrite the exponents as radicals:
  • $a^{5/18} = \sqrt[18]{a^5}$,
  • $b^{1/6} = \sqrt[6]{b}$.

Thus, the final expression in radical notation is:

[ \sqrt[18]{a^5} \cdot \sqrt[6]{b}. ]

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Would you like a step-by-step verification or details on any specific part? Here are related questions:

1. How do you simplify expressions with different radicals like this?
2. What are the rules for combining fractional exponents?
3. How do you convert between exponents and radical notation?
4. Why is finding a common denominator necessary when subtracting exponents?
5. Can you explain more about the assumptions for positive variables in this context?

Tip: Always simplify fractional exponents by finding the least common denominator; this ensures the process is systematic and accurate.