To simplify the expression $\frac{3(x^2 + 1)^{\frac{2}{3}}}{(x^2 + 1)^{-\frac{1}{3}}}$, you can use the rules of exponents.

Step 1: Apply the Quotient Rule of Exponents

The quotient rule of exponents states that when you divide two expressions with the same base, you subtract the exponents:

$\frac{a^m}{a^n} = a^{m-n}$

Applying this to the given expression:

$\frac{3(x^2 + 1)^{\frac{2}{3}}}{(x^2 + 1)^{-\frac{1}{3}}} = 3(x^2 + 1)^{\frac{2}{3} - \left(-\frac{1}{3}\right)}$

Step 2: Simplify the Exponents

Subtract the exponents:

$\frac{2}{3} - \left(-\frac{1}{3}\right) = \frac{2}{3} + \frac{1}{3} = \frac{3}{3} = 1$

So the expression simplifies to:

$3(x^2 + 1)^1$

Final Answer

The simplified expression is:

$3(x^2 + 1)$

---

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How do you simplify expressions involving negative exponents?
2. What is the general rule for multiplying and dividing expressions with the same base?
3. How would the expression change if the exponents were fractions other than $\frac{2}{3}$ and $-\frac{1}{3}$?
4. What are some common mistakes to avoid when working with exponents?
5. Can you factor or expand the expression $3(x^2 + 1)$ further?

Tip: When dividing expressions with the same base, always remember to subtract the exponents. This makes simplification much easier!