(41C) Exponentiation of Large Numbers

[Eddie W. Shore]

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

Code:


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

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

Blog post: http://edspi31415.blogspot.com/2019/08/h...on-of.html