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?

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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.