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Let's break down the given expression and simplify it step by step.

The expression is:

\[
\frac{\sqrt[6]{a^5}}{\sqrt{a} : \sqrt[3]{a}}
\]

### Step 1: Simplify the denominator

The expression in the denominator is \(\sqrt{a} : \sqrt[3]{a}\). The colon (:) denotes division, so this part is equivalent to:

\[
\frac{\sqrt{a}}{\sqrt[3]{a}} = \frac{a^{1/2}}{a^{1/3}}.
\]

To divide powers with the same base, subtract the exponents:

\[
a^{1/2 - 1/3} = a^{3/6 - 2/6} = a^{1/6}.
\]

So the denominator simplifies to \(a^{1/6}\).

### Step 2: Simplify the numerator

The numerator is \(\sqrt[6]{a^5}\), which can be written as:

\[
a^{5/6}.
\]

### Step 3: Simplify the entire expression

Now, the expression becomes:

\[
\frac{a^{5/6}}{a^{1/6}}.
\]

To simplify, subtract the exponents:

\[
a^{5/6 - 1/6} = a^{4/6} = a^{2/3}.
\]

### Final Answer:

The simplified expression is:

\[
a^{2/3}.
\]

Would you like more details or explanations on any step? Here are some related questions:

1. What are the general rules for simplifying exponents with the same base?
2. How does dividing powers of the same base work?
3. What are the steps for converting radical expressions to fractional exponents?
4. How do you simplify an expression like \( a^{m/n} \)?
5. Can we extend these exponent rules to negative or zero exponents?

**Tip:** Remember that radicals can always be converted into fractional exponents, which simplifies operations involving them!

Simplify the expression: \(\frac{\sqrt[6]{a^5}}{\sqrt{a} : \sqrt[3]{a}}\)

Learn how to simplify a complex expression involving radicals and fractional exponents. This detailed solution walks through dividing radicals, converting them to fractional exponents, and applying exponent rules. Suitable for high school math students.

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