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.

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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.