Let's work through the problem shown in the image step by step.

Problem:

You are asked to simplify and write the expression $(5a^{4/5})(4a^{4/5})$ in radical notation.

Steps to Solve:

1. Combine the constants: $5 \times 4 = 20$ So, the constants become $20$.

2. Combine the powers of $a$: Both terms have $a$ raised to the $\frac{4}{5}$-th power, so you add the exponents: $a^{4/5} \times a^{4/5} = a^{(4/5 + 4/5)} = a^{8/5}$

3. Convert to radical notation: The exponent $a^{8/5}$ can be rewritten as: $a^{8/5} = \sqrt[5]{a^8}$ Therefore, the expression becomes: $20 \cdot \sqrt[5]{a^8}$

Answer:

From the choices given, the correct simplified form is:

$\boxed{C. \ 20 \sqrt[5]{a^8}}$

Would you like more details on any part of this? Feel free to ask!

Follow-up Questions:

1. How would you simplify $(3a^{2/3})(7a^{1/3})$?
2. What is the general rule for multiplying terms with the same base but different exponents?
3. How would you simplify $(6b^{1/2})(4b^{3/4})$?
4. Can you explain why the expression $a^{m/n}$ can be written in radical notation?
5. What happens if you subtract exponents when dividing terms with the same base?

Tip:

When combining exponents with the same base, always remember to add the exponents for multiplication and subtract them for division.