The expression for $a_n$ provided in the image is:

[ a_n = \frac{\sqrt[4]{3n^6} - \sqrt{4n^5} + 3n}{\sqrt[3]{9n^{10}} + 5n^4 - 4n^2} ]

Let me simplify it step by step.

1. Numerator:

$\sqrt[4]{3n^6} - \sqrt{4n^5} + 3n$

• $\sqrt[4]{3n^6} = (3n^6)^{1/4} = 3^{1/4} n^{6/4} = 3^{1/4} n^{3/2}$
• $\sqrt{4n^5} = (4n^5)^{1/2} = 2n^{5/2}$

Thus, the numerator becomes: $3^{1/4} n^{3/2} - 2n^{5/2} + 3n$

2. Denominator:

$\sqrt[3]{9n^{10}} + 5n^4 - 4n^2$

• $\sqrt[3]{9n^{10}} = (9n^{10})^{1/3} = 9^{1/3} n^{10/3}$

Thus, the denominator becomes: $9^{1/3} n^{10/3} + 5n^4 - 4n^2$

Final Simplified Expression:

The final simplified expression for $a_n$ is:

$a_n = \frac{3^{1/4} n^{3/2} - 2n^{5/2} + 3n}{9^{1/3} n^{10/3} + 5n^4 - 4n^2}$

Would you like further details or have any questions about the simplification?

Here are 5 related questions to deepen understanding:

1. What is the significance of fractional exponents in simplifying algebraic expressions?
2. How can we compare growth rates of terms like $n^{5/2}$ and $n^4$?
3. What happens to the expression as $n \to 0$?
4. How would the expression change if the powers of $n$ in the denominator were reduced?
5. Can we approximate $a_n$ for large $n$?

Tip: When dealing with fractional exponents, remember that they represent roots, which can help in factoring or simplifying complex terms.