The image shows two problems for simplification:

Problem 1:

$\frac{a^{\frac{1}{3}} \times a^{\frac{2}{3}}}{a^{\frac{1}{3}}}$

Problem 2:

$(2xy^3)^4$

Let's solve both of them:

For the first problem:

$\frac{a^{\frac{1}{3}} \times a^{\frac{2}{3}}}{a^{\frac{1}{3}}}$

1. Apply the product rule for exponents to simplify the numerator: $a^{\frac{1}{3}} \times a^{\frac{2}{3}} = a^{\frac{1}{3} + \frac{2}{3}} = a^1 = a$

2. Now, we have: $\frac{a}{a^{\frac{1}{3}}}$

3. Apply the quotient rule for exponents: $\frac{a^1}{a^{\frac{1}{3}}} = a^{1 - \frac{1}{3}} = a^{\frac{2}{3}}$

So, the simplified form of the first expression is: $a^{\frac{2}{3}}$

For the second problem:

$(2xy^3)^4$

1. Apply the power rule to each factor inside the parentheses: $(2)^4 \times (x)^4 \times (y^3)^4$

2. Simplify each part: $2^4 = 16, \quad x^4 = x^4, \quad (y^3)^4 = y^{12}$

3. So, the simplified form is: $16x^4y^{12}$

Final Answers:

1. $a^{\frac{2}{3}}$
2. $16x^4y^{12}$

Let me know if you'd like further clarification or have any other questions!

Here are 5 follow-up questions related to the concepts covered:

1. How do the product and quotient rules for exponents work with fractional exponents?
2. What is the general power rule for exponents when dealing with powers of products?
3. How do you simplify expressions involving negative exponents?
4. What happens if you have different base numbers in a similar expression? How do you simplify it?
5. How can you simplify expressions involving square roots or cube roots using exponents?

Tip: Always check that your base is the same before applying the product and quotient rules!