Convert the following:
(i) (562.13)7 = (?)10
(ii) (467.342)10 =(?)5
 (iii (AE29.11)16 = (?)8

Let's convert each of the given numbers to the desired base step by step.

### (i) \((562.13)_7 = (?)_{10}\)
To convert a number from base 7 to base 10, we need to expand it in powers of 7.

\[
(562.13)_7 = 5 \times 7^2 + 6 \times 7^1 + 2 \times 7^0 + 1 \times 7^{-1} + 3 \times 7^{-2}
\]

Now calculate each term:
\[
5 \times 49 = 245
\]
\[
6 \times 7 = 42
\]
\[
2 \times 1 = 2
\]
\[
1 \times \frac{1}{7} \approx 0.142857
\]
\[
3 \times \frac{1}{49} \approx 0.061224
\]

Adding them up:
\[
245 + 42 + 2 + 0.142857 + 0.061224 \approx 289.204081
\]

So, 
\[
(562.13)_7 = (289.204081)_{10}
\]

### (ii) \((467.342)_{10} = (?)_5\)

To convert a decimal number from base 10 to base 5, we need to convert both the integer and the fractional parts separately.

1. **Integer part** \(467_{10}\):

   - Divide 467 by 5, quotient = 93, remainder = 2
   - Divide 93 by 5, quotient = 18, remainder = 3
   - Divide 18 by 5, quotient = 3, remainder = 3
   - Divide 3 by 5, quotient = 0, remainder = 3
   
   So, \(467_{10} = (3332)_5\).

2. **Fractional part** \(0.342_{10}\):

   - Multiply 0.342 by 5, result = 1.71, integer part = 1
   - Multiply 0.71 by 5, result = 3.55, integer part = 3
   - Multiply 0.55 by 5, result = 2.75, integer part = 2
   - Multiply 0.75 by 5, result = 3.75, integer part = 3 (This can continue, but we'll stop here for precision.)

   So, \(0.342_{10} \approx (0.1323)_5\).

Combining both parts:
\[
(467.342)_{10} \approx (3332.1323)_5
\]

### (iii) \((AE29.11)_{16} = (?)_8\)
To convert from base 16 to base 8, we can first convert the hexadecimal number to binary, and then from binary to octal.

1. **Convert to binary:**

   - \(A_{16} = 1010_2\)
   - \(E_{16} = 1110_2\)
   - \(2_{16} = 0010_2\)
   - \(9_{16} = 1001_2\)
   - \(1_{16} = 0001_2\)
   
   So,
   \[
   (AE29.11)_{16} = (1010\ 1110\ 0010\ 1001.0001\ 0001)_2
   \]

2. **Convert to octal** (group binary digits in sets of 3):

   - Group the binary digits from right to left:
     \[
     101\ 011\ 100\ 010\ 100.001\ 000\ 100
     \]
     
   - Convert each group to octal:
     \[
     101_2 = 5_8, \ 011_2 = 3_8, \ 100_2 = 4_8, \ 010_2 = 2_8, \ 100_2 = 4_8
     \]
     \[
     001_2 = 1_8, \ 000_2 = 0_8, \ 100_2 = 4_8
     \]

   So,
   \[
   (AE29.11)_{16} = (53424.104)_8
   \]

### Final Answers:
1. \((562.13)_7 \approx (289.204081)_{10}\)
2. \((467.342)_{10} \approx (3332.1323)_5\)
3. \((AE29.11)_{16} = (53424.104)_8\)

Would you like more details or have any questions?

---

Here are 5 related questions for further practice:
1. Convert \((102.4)_8\) to base 10.
2. Convert \((1100101)_2\) to base 10.
3. Convert \((453.21)_6\) to base 7.
4. Convert \((9345.67)_{10}\) to base 2.
5. Convert \((ABC.123)_{16}\) to base 10.

**Tip:** When converting between bases, converting through base 2 (binary) can often simplify the process, especially for bases that are powers of 2, like base 4, 8, or 16.

Learn how to convert numbers between different bases, including base 7 to base 10, base 10 to base 5, and base 16 to base 8. Detailed step-by-step solutions provided.

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