The problem asks to express the given expression in terms of roots with a base of $a$, where $a \in \mathbb{N}$.

The expression provided is:

$a^{\frac{1}{2}} : \sqrt[6]{a}$

To solve this, let’s analyze and simplify step by step.

1. Rewrite $a^{\frac{1}{2}}$: This is already in the form of a power, and it represents the square root of $a$: $a^{\frac{1}{2}} = \sqrt{a}$

2. Rewrite $\sqrt[6]{a}$: This is the sixth root of $a$, which can be written as a power: $\sqrt[6]{a} = a^{\frac{1}{6}}$

3. Formulate the division: Now, we have $a^{\frac{1}{2}} : a^{\frac{1}{6}} = a^{\frac{1}{2} - \frac{1}{6}}$

4. Simplify the exponent: $\frac{1}{2} - \frac{1}{6} = \frac{3 - 1}{6} = \frac{2}{6} = \frac{1}{3}$

Thus, the expression simplifies to:

$a^{\frac{1}{3}}$

5. Express in root form: $a^{\frac{1}{3}}$ can be rewritten as the cube root of $a$: $a^{\frac{1}{3}} = \sqrt[3]{a}$

Final Answer

[ a^{\frac{1}{2}} : \sqrt[6]{a} = \sqrt[3]{a} ]

Would you like further details on any part of this solution, or do you have any questions?

Here are some related questions for further exploration:

1. What are the rules for simplifying expressions with roots and powers?
2. How can we express higher roots, like the sixth root, in terms of fractional exponents?
3. What other methods can be used to simplify expressions involving roots and powers?
4. How does changing the base of the exponent affect the result of a similar expression?
5. What practical applications involve simplifying expressions with roots and fractional exponents?

Tip: When simplifying expressions with powers and roots, remember that division of terms with the same base allows you to subtract the exponents directly.