## Monday, August 19, 2019

### HP 41/DM41L and TI-60X: Exponentiation of Large Numbers

HP 41/DM41L  and TI-60X:  Exponentiation of Large Numbers

But Why a Program when we have Button?

This is true.  What this program does is allow for calculation of y^x when results in answers greater than 9.999999999 * 10^9.  The number is broken up into the form:

mantissa * 10^exponent

Let n = y^x.  Then:

n = y^x

Taking the logarithm of both sides:

log n = log (y^x)
log n = x log y

A number can be split into its fractional and integer part:

log n = frac(x log y) + int(x log y)

Take the antilog of both sides:

n = 10^( frac(x log y) + int(x log y) )
n = 10^( frac(x log y) ) * 10^( int(x log y) )

where
mantissa = 10^( frac(x log y) )
exponent = int(x log y)

HP 41/DM 41L Program BIGPOW

Input:
Y stack:  y
X stack:  x

Output:
Y:  mantissa (shown first)
X:  exponent

01 LBL T^BIGPOW
02 X<>Y
03 LOG
04 *
05 ENTER↑
06 FRC
07 10↑X
08 STOP
09 X<>Y
10 INT
11 RTN

TI-60 Program:  Big Powers

Input:
Store y in R1 and x in R2

Output:
R1 = mantissa (shown first), R2 = exponent

(Step,  Key Number, Key)
00, 71, RCL
01, 02, 2
02, 65, *
03, 71, RCL
04, 01, 1
05, 43, log
06, 95, =
07, 61, STO
08, 02, 2
09, 78, Frac
10, 12, INV
11, 43, log
12, 61, STO
13, 01, 1
14, 13, R/S
15, 71, RCL
16, 02, 2
17, 79, Intg
18, 13, R/S
19, 22, RST

Examples

Example 1:  25^76.   y = 25, x = 76

Result:
Mantissa = 1.75162308
Exponent = 106
25^76 ≈ 1.75162308 * 10^106

Example 2:  78^55.25,  y = 78, x = 55.25

Result:
Mantissa = 3.453240284
Exponent = 104
78^55.25 ≈ 3.543240284 * 10^104

Eddie

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