Consider the real valued variables $X$, $Y$ and $Z$ represented using the IEEE 754 single-precision floating-point format. The binary representations of $X$ and $Y$ in hexadecimal notation are as follows: $$X\text{: } 35\text{C}00000 \qquad Y\text{: } 34\text{A}00000$$ Let $Z = X + Y$. Which one of the following is the binary representation of $Z$, in hexadecimal notation?

In the IEEE 754 single-precision floating-point format (32 bits):

• Sign ($S$): 1 bit

• Biased Exponent ($E$): 8 bits (Bias = 127)

• Fraction / Mantissa ($M$): 23 bits (with an implicit leading 1)

1. Decode $X = 35\text{C}00000_{16}$:

• Binary: 0011 0101 1100 0000 0000 0000 0000 0000

• Sign bit $S_X = 0$ (positive)

• Exponent bits $E_X = 01101011_2 = 107$

• Mantissa bits $M_X = 100\ 0000\dots_2$

• Significand: $1.1_2$

• Value: $X = +1.1_2 \times 2^{107 - 127} = 1.1_2 \times 2^{-20}$

2. Decode $Y = 34\text{A}00000_{16}$:

• Binary: 0011 0100 1010 0000 0000 0000 0000 0000

• Sign bit $S_Y = 0$ (positive)

• Exponent bits $E_Y = 01101001_2 = 105$

• Mantissa bits $M_Y = 010\ 0000\dots_2$

• Significand: $1.01_2$

• Value: $Y = +1.01_2 \times 2^{105 - 127} = 1.01_2 \times 2^{-22}$

3. Addition $Z = X + Y$:

• Align exponents to the larger exponent ($E = 107$, difference $\Delta E = 107 - 105 = 2$):

  $$Y = 0.0101_2 \times 2^{-20}$$

• Add the significands:

  $$\text{Significand of } Z = 1.1000_2 + 0.0101_2 = 1.1101_2$$

• Result: $Z = 1.1101_2 \times 2^{-20}$

4. Encode $Z$ into IEEE 754 format:

• Sign bit $S_Z = 0$

• Biased Exponent $E_Z = 107 = 01101011_2$

• Fraction $M_Z = 1101000\dots_2$

• 32-bit binary pattern: 0 01101011 11010000000000000000000

• Grouping into 4-bit nibbles: 0011 0101 1110 1000 0000 0000 0000 0000

• Hexadecimal: $35\text{E}80000_{16}$

Thus, option (C) is the correct representation.