Question 45

Decode $P = \text{0xC1800000}$:

• Binary: $1\; 10000011\; 00000000000000000000000_2$
• Sign bit $S = 1$ (negative)
• Exponent: $10000011_2 = 131 \implies E_P = 131 - 127 = 4$
• Mantissa: $1.0_2$
• Value: $P = -1.0 \times 2^4 = -16$

Multiply $P \times Q$:

• $P \times Q = -16 \times Q = -2^4 \times Q$
• Since $Q = \text{0x3F5C2EF4}$ is positive (sign bit 0), the product has sign bit $S = 1$.
• Multiplying by $2^4$ simply increases the biased exponent of $Q$ by $4$, keeping the significand identical!
• Biased exponent of $Q$: $0x3F5\dots \implies 01111110_2 = 126$.
• New biased exponent: $126 + 4 = 130 = 10000010_2$.
• Significand remains $5C2EF4_{16}$.

Encode the result:

• Sign: $1$
• Exponent: $10000010$
• Fraction: $10111000010111011110100$
• In hex: $\text{0xC15C2EF4}$.