To convert the number \(562741_8\) (base 8) to base 2, we can follow these steps:

### Step 1: Convert each base-8 digit to its 3-digit binary equivalent.
In base 8, each digit represents 3 binary digits (since \(8 = 2^3\)):

- \(5_8 = 101_2\)
- \(6_8 = 110_2\)
- \(2_8 = 010_2\)
- \(7_8 = 111_2\)
- \(4_8 = 100_2\)
- \(1_8 = 001_2\)

### Step 2: Write down the binary equivalents in sequence.
Now, place the binary representations of the digits next to each other:

\[
5_8 = 101_2,\ 6_8 = 110_2,\ 2_8 = 010_2,\ 7_8 = 111_2,\ 4_8 = 100_2,\ 1_8 = 001_2
\]

Thus, \(562741_8\) in binary is:

\[
101\ 110\ 010\ 111\ 100\ 001_2
\]

### Final Answer:
\[
562741_8 = 101110010111100001_2
\]

Would you like further details or have any questions?

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Here are five related questions for further exploration:
1. How would you convert a number from base 8 to base 10?
2. What is the base 16 equivalent of \(562741_8\)?
3. How can you convert \(101110010111100001_2\) back to base 8?
4. What is the general method for converting between any two bases?
5. How would you add two base 8 numbers directly?

**Tip:** When converting between bases, remember that grouping or expanding digits (like 3 binary digits per octal digit) can simplify conversions.

Learn how to convert the octal number 562741 to binary. This detailed guide covers base conversion concepts, including how each base-8 digit translates into 3 binary digits, providing a thorough understanding for students in Grades 8-10.

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