The given problem asks to simplify the expression:

$\left( a^2 \right)^{\frac{1}{2}} \cdot \left( \sqrt[3]{a} \right)^6$

Step-by-step solution:

1. Simplify the first term:

$\left( a^2 \right)^{\frac{1}{2}} = a^{2 \cdot \frac{1}{2}} = a^1 = a$

2. Rewrite the second term using fractional exponents:

$\sqrt[3]{a} = a^{\frac{1}{3}}$ So: $\left( \sqrt[3]{a} \right)^6 = \left( a^{\frac{1}{3}} \right)^6 = a^{\frac{1}{3} \cdot 6} = a^2$

3. Combine the results:

The original expression now becomes: $a \cdot a^2 = a^{1 + 2} = a^3$

Final Answer:

$\boxed{A.\ a^3}$

Let me know if you would like further clarification! Below are additional related questions for practice.

5 Related Questions:

1. How do you simplify expressions with fractional exponents like $(a^m)^n$?
2. What is the rule for multiplying terms with the same base (e.g., $a^m \cdot a^n$)?
3. How do you convert roots (e.g., square root or cube root) into fractional exponents?
4. What happens when you raise a power to another power in exponents?
5. Can $a^0$ ever be a valid expression, and what is its value?

Tip:

Always remember that roots can be written as fractional exponents. For example: $\sqrt[n]{a^m} = a^{m/n}$