The expression in the image is:

[ \sqrt[3]{a} \times \sqrt[4]{a^3} ]

Let's simplify this step by step.

Step 1: Express both roots as powers of $a$

• The cube root of $a$ is $a^{1/3}$.
• The fourth root of $a^3$ is $a^{3/4}$.

So, the expression becomes: $a^{1/3} \times a^{3/4}$

Step 2: Apply the law of exponents

When multiplying terms with the same base, we add the exponents. Thus: $a^{1/3 + 3/4}$

Step 3: Add the exponents

To add $1/3$ and $3/4$, find a common denominator. The least common denominator between 3 and 4 is 12: $\frac{1}{3} = \frac{4}{12}, \quad \frac{3}{4} = \frac{9}{12}$ Now, add them: $\frac{4}{12} + \frac{9}{12} = \frac{13}{12}$

Thus, the expression becomes: $a^{13/12}$

Final Answer:

The simplified expression is: $a^{13/12}$

Would you like more details on any of the steps, or do you have any further questions?

Here are five related questions for further practice:

1. How would you simplify $\sqrt[5]{a^2} \times \sqrt[2]{a^3}$?
2. What is the general rule for multiplying radical expressions?
3. Can you explain how fractional exponents relate to roots?
4. How do you handle different roots (e.g., cube root vs. fourth root) in one expression?
5. How would you simplify the expression $\sqrt[3]{x} \times \sqrt[6]{x^2}$?

Tip: Always convert roots to exponents for easier manipulation when simplifying expressions involving radicals.