https://file.aiquickdraw.com/mathbot/file/17298617017144tb3d4pn.jpg

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.

Simplify the expression a_n = (sqrt[4]{3n^6} - sqrt{4n^5} + 3n) / (sqrt[3]{9n^10} + 5n^4 - 4n^2).

Explore the step-by-step process of simplifying a complex expression involving fourth, square, and cube roots. This algebraic problem helps students master radicals, exponents, and algebraic simplifications. Ideal for Grades 11-12.

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