(37320.43)_base8 in hexadecimal system is1 point(7DA1.18)_base16(3ED0.8C)_base16(7DA1.11)_base16(3ED0.83)_base16

To convert the octal number $(37320.43)_{8}$ to hexadecimal, we will follow a systematic process.

1. Break Down the Problem

We need to:

1. Convert the octal number to decimal.
2. Convert the decimal number to hexadecimal.

2. Relevant Concepts

• Octal to Decimal Conversion: Each digit in an octal number is multiplied by $8^{n}$, where $n$ is the position index starting from $0$ from the right.
• Decimal to Hexadecimal Conversion: Each digit in a decimal number can be converted to hexadecimal using division by $16$ and noting the remainders.

3. Analysis and Detail

Step 1: Convert Octal to Decimal

$(37320.43)_{8} = 3 \cdot 8^4 + 7 \cdot 8^3 + 3 \cdot 8^2 + 2 \cdot 8^1 + 0 \cdot 8^0 + 4 \cdot 8^{-1} + 3 \cdot 8^{-2}$

Calculating each term:

• $3 \cdot 8^4 = 3 \cdot 4096 = 12288$
• $7 \cdot 8^3 = 7 \cdot 512 = 3584$
• $3 \cdot 8^2 = 3 \cdot 64 = 192$
• $2 \cdot 8^1 = 2 \cdot 8 = 16$
• $0 \cdot 8^0 = 0 \cdot 1 = 0$

For the fractional part:

• $4 \cdot 8^{-1} = 4 \cdot \frac{1}{8} = 0.5$
• $3 \cdot 8^{-2} = 3 \cdot \frac{1}{64} = 0.046875$

Adding all together:

$12288 + 3584 + 192 + 16 + 0 + 0.5 + 0.046875 = 15880.546875$

Step 2: Convert Decimal to Hexadecimal

Now, we will convert $15880.546875$ to hexadecimal.

Convert the integer part $15880$:

1. Divide by $16$ and note the remainders:

• $15880 \div 16 = 992$ remainder $8$
• $992 \div 16 = 62$ remainder $0$
• $62 \div 16 = 3$ remainder $14$ (which is $E$)
• $3 \div 16 = 0$ remainder $3$

So, reading from bottom to top, $15880_{10} = 3E08_{16}$.

Convert the fractional part $0.546875$:

1. Multiply by $16$ and note the integer parts:

• $0.546875 \cdot 16 = 8.75$ (integer part $8$)
• $0.75 \cdot 16 = 12.0$ (integer part $12$ which is $C$)

So, $0.546875_{10} = 0.8C_{16}$.

Combining both parts: $15880.546875_{10} = (3E08.8C)_{16}$

4. Verify and Summarize

The conversions were performed using standard methods for both octal to decimal and decimal to hexadecimal, and no errors were noted throughout the calculations.

Final Answer

The hexadecimal representation of $(37320.43)_{8}$ is $(3E08.8C)_{16}$.