Let's go through each part of the question step-by-step and provide detailed solutions:
### 1. Convert the base 8 number 4.72 to base 10
To convert the number from base 8 to base 10, we need to convert both the integer and fractional parts separately, and then combine them.
Step by step:
1. Convert the integer part:
- [tex]\(4_8\)[/tex] means [tex]\(4\)[/tex] in base 8.
- To convert to base 10: [tex]\(4_8 = 4_{10}\)[/tex].
2. Convert the fractional part:
- [tex]\(0.72_8\)[/tex]
- Each digit in the fractional part is divided by a power of 8.
- [tex]\(7 \times 8^{-1} = 7 \times \frac{1}{8} = 0.875\)[/tex].
- [tex]\(2 \times 8^{-2} = 2 \times \frac{1}{64} = 0.03125\)[/tex].
3. Add these values together:
- [tex]\(4 + 0.875 + 0.03125 = 4.90625\)[/tex].
So, [tex]\(4.72\)[/tex] in base 8 is equivalent to [tex]\(4.90625\)[/tex] in base 10.
### 2. Convert the base 4 number 301 to base 10
To convert a number from base 4 to base 10, we break it down by each digit multiplied by 4 raised to the power of its position (0-indexed from right).
Step by step:
- [tex]\(301_4\)[/tex]
- [tex]\(3 \times 4^2 + 0 \times 4^1 + 1 \times 4^0\)[/tex]
- [tex]\(3 \times 16 + 0 \times 4 + 1 \times 1\)[/tex]
- [tex]\(48 + 0 + 1 = 49\)[/tex]
So, [tex]\(301\)[/tex] in base 4 is equivalent to [tex]\(49\)[/tex] in base 10.
### 3. Convert the base 7 number 324 to base 10
To convert a number from base 7 to base 10, we break it down by each digit multiplied by 7 raised to the power of its position (0-indexed from right).
Step by step:
- [tex]\(324_7\)[/tex]
- [tex]\(3 \times 7^2 + 2 \times 7^1 + 4 \times 7^0\)[/tex]
- [tex]\(3 \times 49 + 2 \times 7 + 4 \times 1\)[/tex]
- [tex]\(147 + 14 + 4 = 165\)[/tex]
So, [tex]\(324\)[/tex] in base 7 is equivalent to [tex]\(165\)[/tex] in base 10.
### Summary of Results
1. [tex]\(4.72_8 = 4.90625_{10}\)[/tex]
2. [tex]\(301_4 = 49_{10}\)[/tex]
3. [tex]\(324_7 = 165_{10}\)[/tex]
We have completed the detailed, step-by-step conversion of the given numbers into base 10.
