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?

---

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.