Let's simplify the given expression step by step using rational exponents and convert back to radical notation if necessary. The expression is:

$\sqrt{a^7b^6 \cdot 9\sqrt{a}}$

Step 1: Rewrite the roots as rational exponents

1. The outer square root ($\sqrt{x}$) corresponds to an exponent of $1/2$.
2. The nested $9\sqrt{a}$ corresponds to $9a^{1/2}$.

Thus, the expression becomes:

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

Step 2: Combine the exponents

Inside the square root, simplify the powers of $a$. Adding the exponents:

$a^7 \cdot a^{1/2} = a^{7 + 1/2} = a^{15/2}$

Now the expression becomes:

$\sqrt{a^{15/2} b^6 \cdot 9}$

Step 3: Apply the square root ($1/2$) to the entire expression

Distribute the square root ($1/2$ exponent) to each factor inside:

$\sqrt{a^{15/2} b^6 \cdot 9} = \sqrt{a^{15/2}} \cdot \sqrt{b^6} \cdot \sqrt{9}$

Simplify each term:

1. $\sqrt{a^{15/2}} = a^{15/4}$
2. $\sqrt{b^6} = b^{6/2} = b^3$
3. $\sqrt{9} = 3$

Now the expression is:

$3a^{15/4}b^3$

Step 4: Convert back to radical notation if needed

The exponent $15/4$ can be written as:

$a^{15/4} = \sqrt[4]{a^{15}}$

Thus, the final expression in radical form is:

$3b^3 \cdot \sqrt[4]{a^{15}}$

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Final Answer:

$3b^3 \cdot \sqrt[4]{a^{15}}$

Would you like further explanation or examples? 😊

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5 Related Questions:

1. How do rational exponents and roots relate in algebraic expressions?
2. Can you simplify a similar expression involving cube roots instead of square roots?
3. How do you handle variables with negative exponents in radical notation?
4. What are the key rules