onvert each of the following numbers to their respective number systems equive a) \( \mathrm{A}^{2} \mathrm{DFE}{ }_{16} \) to binary. b) \( 11110000_{2} \) to octal. c) \( 65_{10} \) to hexadecimal. d) \( 673_{8} \) to decimal.

Let's convert each of the given numbers to their respective number systems. ### a) Convert \( \mathrm{A}^{2} \mathrm{DFE}_{16} \) to binary. To convert from hexadecimal (base 16) to binary (base 2), we can convert each hexadecimal digit to its 4-bit binary equivalent. - \( \mathrm{A} = 1010 \) - \( 2 = 0010 \) - \( D = 1101 \) - \( F = 1111 \) - \( E = 1110 \) Now, we can combine these binary representations: \[ \mathrm{A}^{2} \mathrm{DFE}_{16} = 1010 \, 0010 \, 1101 \, 1111 \, 1110_{2} \] So, \( \mathrm{A}^{2} \mathrm{DFE}_{16} \) in binary is: \[ 10100010110111111110_{2} \] ### b) Convert \( 11110000_{2} \) to octal. To convert from binary (base 2) to octal (base 8), we group the binary digits into sets of three, starting from the right. If necessary, we can add leading zeros. Grouping \( 11110000 \): \[ \text{Add leading zeros: } 001 \, 111 \, 000 \] Now convert each group to octal: - \( 001 = 1 \) - \( 111 = 7 \) - \( 000 = 0 \) Combining these gives: \[ 11110000_{2} = 170_{8} \] ### c) Convert \( 65_{10} \) to hexadecimal. To convert from decimal (base 10) to hexadecimal (base 16), we can repeatedly divide the number by 16 and keep track of the remainders. 1. \( 65 \div 16 = 4 \) remainder \( 1 \) 2. \( 4 \div 16 = 0 \) remainder \( 4 \) Reading the remainders from bottom to top, we get: \[ 65_{10} = 41_{16} \] ### d) Convert \( 673_{8} \) to decimal. To convert from octal (base 8) to decimal (base 10), we can expand the number using powers of 8. \[ 673_{8} = 6 \times 8^2 + 7 \times 8^1 + 3 \times 8^0 \] Calculating each term: - \( 6 \times 8^2 = 6 \times 64 = 384 \) - \( 7 \times 8^1 = 7 \times 8 = 56 \) - \( 3 \times 8^0 = 3 \times 1 = 3 \) Now, summing these values: \[ 384 + 56 + 3 = 443 \] So, \( 673_{8} = 443_{10} \). ### Summary of Results: a) \( \mathrm{A}^{2} \mathrm{DFE}_{16} = 10100010110111111110_{2} \) b) \( 11110000_{2} = 170_{8} \) c) \( 65_{10} = 41_{16} \) d) \( 673_{8} = 443_{10} \)