Divide the following numbers by the given index radical: 1. [9]1009 2. [4]828 3. [13]525 4. [17]684 5. [41]1331 6. [23]713

Divide 12 by 9.

$$\begin{array}{r} 11 \\ 9 \overline{) 1029} \\ -9 \downarrow \\ \hline 12 \\ -9 \downarrow \\ \hline 3 \end{array}$$

Divide 39 by 9.

$$\begin{array}{r} 114 \\ 9 \overline{) 1029} \\ -9 \downarrow \\ \hline 12 \\ -9 \downarrow \\ \hline 39 \\ -36 \\ \hline 3 \end{array}$$

The quotient is 114 with a remainder of 3.
So, $\frac{1029}{9} = 114 \text{ R } 3$.

Final Answer: $\boxed{114 \text{ R } 3}$

2. Divide 828 by 4

Divide 2 by 4.

$$\begin{array}{r} 20 \\ 4 \overline{) 828} \\ -8 \downarrow \\ \hline 02 \\ -0 \downarrow \\ \hline 2 \end{array}$$

Divide 28 by 4.

$$\begin{array}{r} 207 \\ 4 \overline{) 828} \\ -8 \downarrow \\ \hline 02 \\ -0 \downarrow \\ \hline 28 \\ -28 \\ \hline 0 \end{array}$$

The quotient is 207 with a remainder of 0.
So, $\frac{828}{4} = 207$.

Final Answer: $\boxed{207}$

3. Divide 585 by 13

($13 \times 4 = 52$)

$$\begin{array}{r} 4 \\ 13 \overline{) 585} \\ -52 \downarrow \\ \hline 6 \end{array}$$

Divide 65 by 13. ($13 \times 5 = 65$)

$$\begin{array}{r} 45 \\ 13 \overline{) 585} \\ -52 \downarrow \\ \hline 65 \\ -65 \\ \hline 0 \end{array}$$

The quotient is 45 with a remainder of 0.
So, $\frac{585}{13} = 45$.

Final Answer: $\boxed{45}$

4. Divide 684 by 17

($17 \times 4 = 68$)

$$\begin{array}{r} 4 \\ 17 \overline{) 684} \\ -68 \downarrow \\ \hline 0 \end{array}$$

Divide 4 by 17.

$$\begin{array}{r} 40 \\ 17 \overline{) 684} \\ -68 \downarrow \\ \hline 04 \\ -0 \\ \hline 4 \end{array}$$

The quotient is 40 with a remainder of 4.
So, $\frac{684}{17} = 40 \text{ R } 4$.

Final Answer: $\boxed{40 \text{ R } 4}$

5. Divide 1881 by 41

($41 \times 4 = 164$)

$$\begin{array}{r} 4 \\ 41 \overline{) 1881} \\ -164 \downarrow \\ \hline 24 \end{array}$$

Divide 241 by 41. ($41 \times 5 = 205$, $41 \times 6 = 246$)

$$\begin{array}{r} 45 \\ 41 \overline{) 1881} \\ -164 \downarrow \\ \hline 241 \\ -205 \\ \hline 36 \end{array}$$

The quotient is 45 with a remainder of 36.
So, $\frac{1881}{41} = 45 \text{ R } 36$.

Final Answer: $\boxed{45 \text{ R } 36}$

6. Divide 713 by 23

($23 \times 3 = 69$)

$$\begin{array}{r} 3 \\ 23 \overline{) 713} \\ -69 \downarrow \\ \hline 2 \end{array}$$

Divide 23 by 23. ($23 \times 1 = 23$)

$$\begin{array}{r} 31 \\ 23 \overline{) 713} \\ -69 \downarrow \\ \hline 23 \\ -23 \\ \hline 0 \end{array}$$

The quotient is 31 with a remainder of 0.
So, $\frac{713}{23} = 31$.

Final Answer: $\boxed{31}$