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A $32$-bit floating-point number is represented by a $7$-bit signed exponent, and a $24$-bit fractional mantissa. The base of the scale factor is $16,$
The range of the exponent is ___________, if the scale factor is represented in excess-$64$ format.
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Minimum number $= 0$ , maximum number $= 127$,

Given excess $64$ so, bias number is $64$,

Range will be $0-64=-64$ to $(127-64=63) 63$.

Here, in question given that base is $16$, so the actual value represented will be $(-1)^s ( 0.M) \times16 ^{E-\text{bias}}.$
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S Exponent Mantissa

31        30                                                            23                                                         0

The number is represented as (-1)S x 0.Mantissa x 10Exp - 64

a) Range of the exponent is range of 7-bit signed numbers, i.e -64 to 63. If you have confusion in range of signed integers, refer http://kias.dyndns.org/comath/13.html 

b) In excess-k format, actual exponent = Exp - 64

Exp - 64 = -64  and Exp - 64 = +63

So actual range of Exponent is 0 to +127   

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It is asked , range of the exponent in excess 64 format , i.e. double precession format

Single Precession: SEEEEEEE EMMMMMMM MMMMMMMM MMMMMMMM
Double Precession: SEEEEEEE EEEEMMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM

Now for positive or negative normalize real number exponent range must be in between 000.....01  to 11.....10

Now for single precession it will be 0000 0001 to 1111 1110

and in double precession it will be 0000 0000 001 to 1111 1111 110

http://steve.hollasch.net/cgindex/coding/ieeefloat.html

https://gateoverflow.in/194521/dl-ieee-754-single-precision-range

https://gateoverflow.in/82469/coa-floating-point-arithmatic

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