The given expression is:

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

Simplification:

1. Convert the square root and fourth root into exponential form: $\sqrt{a} = a^{1/2}, \quad \sqrt[4]{a^3} = a^{3/4}.$

2. Multiply the terms using the rule of exponents $a^m \cdot a^n = a^{m+n}$: $a^{1/2} \cdot a^{3/4} = a^{1/2 + 3/4}.$

3. Add the exponents: $\frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4}.$

4. Write the final result in simplified form: $a^{5/4}.$

Alternatively, you can express it as: $\sqrt[4]{a^5}.$

Let me know if you'd like further clarification or examples! Here are 5 related questions to expand your understanding:

1. What is the general rule for multiplying terms with fractional exponents?
2. How do you simplify $\sqrt[3]{a^2} \cdot \sqrt{a^3}$?
3. How can $a^{5/4}$ be rewritten as a product of a root and a power?
4. What happens when you multiply $\sqrt[4]{a^3}$ by itself?
5. How do you simplify $\sqrt[6]{a^4} \cdot \sqrt[3]{a^5}$?

Tip: When working with fractional exponents, always aim to express all terms with the same base and consistent fractional powers for easier manipulation.