The question asks to find the value of $P$, given that:

$P_8 = 1010110101_2$

This notation means that $P_8$ is the base-8 representation of the binary number $1010110101_2$. To solve this, we need to first convert the binary number $1010110101_2$ to decimal, and then interpret that as an octal (base-8) number.

Step 1: Convert binary $1010110101_2$ to decimal.

The binary number $1010110101_2$ can be expanded as:

$1010110101_2 = 1 \cdot 2^9 + 0 \cdot 2^8 + 1 \cdot 2^7 + 0 \cdot 2^6 + 1 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0$

Now, compute the powers of 2 and add them:

$= 512 + 0 + 128 + 0 + 32 + 16 + 0 + 4 + 0 + 1 = 693$

So, $1010110101_2$ in decimal is 693.

Step 2: Interpret 693 as a base-8 number $P_8$.

Now, we need to convert the decimal number 693 from base-8 to base-10.

The base-8 number 693 can be expanded as:

$693_8 = 6 \cdot 8^2 + 9 \cdot 8^1 + 3 \cdot 8^0$

Calculating each term:

$6 \cdot 8^2 = 6 \cdot 64 = 384$ $9 \cdot 8^1 = 9 \cdot 8 = 72$ $3 \cdot 8^0 = 3 \cdot 1 = 3$

Adding these together:

$384 + 72 + 3 = 459$

Thus, $P = 459$.

However, none of the given answer options (A: 1265, B: 1625, C: 5231, D: 5321) match this result. Please double-check the details provided in the image or let me know if further clarification is needed!

Would you like to review any other part of this solution or expand on it?

Further Exploration:

1. Can you verify if the binary number was correctly written down?
2. Would you like to explore more examples of binary to octal conversions?
3. Do you need help with another base conversion problem?
4. Do you need more practice converting base-8 numbers to decimal?
5. Are there any other steps you would like me to explain in more detail?

Tip: When converting between number bases, always break the problem into smaller steps, converting one base to decimal before moving on to the final base.