To simplify the expression \( \frac{5a^6}{(a^3\sqrt{a})^3} \), let's break it down step by step.

### Step 1: Rewrite \( \sqrt{a} \) as \( a^{1/2} \)
Since \( \sqrt{a} \) is the same as \( a^{1/2} \), we can rewrite the expression as follows:

\[
\frac{5a^6}{(a^3a^{1/2})^3}
\]

### Step 2: Simplify the base inside the parentheses
When multiplying powers with the same base, add the exponents. So,

\[
a^3a^{1/2} = a^{3 + 1/2} = a^{7/2}
\]

Thus, the expression becomes:

\[
\frac{5a^6}{(a^{7/2})^3}
\]

### Step 3: Apply the exponent to the base
Now, apply the power of 3 to \( a^{7/2} \):

\[
(a^{7/2})^3 = a^{(7/2) \times 3} = a^{21/2}
\]

The expression now looks like:

\[
\frac{5a^6}{a^{21/2}}
\]

### Step 4: Simplify the division of exponents
When dividing powers with the same base, subtract the exponents:

\[
a^6 = a^{12/2}, \quad \text{so} \quad \frac{a^{12/2}}{a^{21/2}} = a^{\frac{12}{2} - \frac{21}{2}} = a^{-\frac{9}{2}}
\]

### Step 5: Final simplified form
Now, the expression is:

\[
5a^{-\frac{9}{2}} = \frac{5}{a^{9/2}} = \frac{5}{a^{4.5}}
\]

Thus, the simplified expression is:

\[
\boxed{\frac{5}{a^{4.5}}}
\]

Would you like more details or have any questions about these steps?

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Here are 5 related questions to consider:
1. How do you simplify expressions with fractional exponents?
2. What is the rule for multiplying powers with the same base?
3. How do you handle negative exponents in expressions?
4. What are the properties of square roots and fractional powers?
5. How do you simplify expressions involving higher powers of radicals?

**Tip:** Always rewrite roots in exponential form when simplifying complex expressions; this makes applying the power rules easier.

Learn how to simplify the expression 5a^6 / (a^3√a)^3 using properties of exponents and fractional powers. This step-by-step solution is ideal for students in Grades 9-12 and covers key algebraic concepts like the power rule and simplifying radicals.

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