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Simplify the radical expression assume that all variable represent positive quantities. Write your answer using radical notation is Nessus

The expression to simplify is:

\[
\frac{\sqrt{24a^6b^7}}{\sqrt{8a^7b^4}}
\]

### Step-by-Step Simplification:

1. **Simplify the Radicals:**
   - First, simplify the expression under the square root for both the numerator and the denominator.

   - **Numerator:**
     \[
     \sqrt{24a^6b^7}
     \]
     - 24 can be factored as \(24 = 4 \times 6\), and \( \sqrt{4} = 2 \).
     - The expression inside the square root becomes:
       \[
       \sqrt{4 \times 6 \times a^6 \times b^7} = 2a^3b^3\sqrt{6b}
       \]

   - **Denominator:**
     \[
     \sqrt{8a^7b^4}
     \]
     - 8 can be factored as \(8 = 4 \times 2\), and \( \sqrt{4} = 2 \).
     - The expression inside the square root becomes:
       \[
       \sqrt{4 \times 2 \times a^7 \times b^4} = 2a^3b^2\sqrt{2ab}
       \]

2. **Simplify the Overall Fraction:**
   - Now, divide the simplified radicals:
     \[
     \frac{2a^3b^3\sqrt{6b}}{2a^3b^2\sqrt{2ab}} 
     \]
     - The constants \(2a^3b^2\) cancel out:
       \[
       \frac{b\sqrt{6b}}{\sqrt{2ab}} 
       \]
     - Combine the square roots:
       \[
       \frac{\sqrt{6b}}{\sqrt{2ab}} = \sqrt{\frac{6b}{2ab}} = \sqrt{\frac{6}{2a}} = \sqrt{\frac{3}{a}}
       \]

### Final Answer:
\[
b \sqrt{\frac{3}{a}}
\]

This is the simplified form of the expression, written in radical notation.

Would you like further details on this solution or have any questions?

### Related Questions:
1. How would you simplify \(\frac{\sqrt{18x^4y^5}}{\sqrt{2x^2y}}\)?
2. What is the simplified form of \(\frac{\sqrt{50m^3n^2}}{\sqrt{2mn}}\)?
3. Can you explain the process of simplifying \(\sqrt{32p^5q^6}\)?
4. How do you simplify expressions involving cube roots, like \(\frac{\sqrt[3]{64x^9y^3}}{\sqrt[3]{8x^3y}}\)?
5. What rules apply when multiplying and dividing square roots with variables?

### Tip:
Always factor the numbers under the square root to identify perfect squares that can be simplified easily.

Learn how to simplify the radical expression \(\frac{\sqrt{24a^6b^7}}{\sqrt{8a^7b^4}}\) step-by-step with a detailed solution. This problem covers radical notation, simplifying fractions involving square roots, and applying properties of exponents. Suitable for students in Grades 9-12.

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