- HELLO!Here is your alternative algorithm for today:
**ALTERNATIVE ALGORITHM # 14****RAISING A NUMBER TO AN INTEGER EXPONENT**The usual method of raising a number to an integer exponent is done by multiplying the number by itself successivelytimes as in*(n-1)**2*^{5}^{ }=x*2*x*2*x*2*x*2*=*2*. This method is manageable if the base number is not more than two digits. But if the base number is more than two digits, the task becomes exasperating. However, there are manual methods for approximating the values of exponentials.*32*a.**THE DIFFERENTIAL EXCESS METHOD**If a number is expressed as a binomial which is raised to exponent, the binomial expansion is:*n**(a*__+__b)^{n}= a^{n}__+__nba^{n-1}+ n(n-1)b^{2}a^{n-2}__+__…b^{n}Ifis very small compared to*b*, the approximate value of the binomial is equal to the sum of the values of the first two terms in the expansion thus:*a**(***a**__+__b)^{n}= a^{n}__+__n ba^{n-1}(5)The value obtained by using Eq.(5) is considered fairly accurate for as long asis at least a two digit integer and*a*is the decimal complement of the given base number.*b***Example:**Find the value of*15.125*^{5}**Solution:**

For the first term;*15.125*^{5}= (15 + 0.125)^{5}= 15^{5}+ 5 x 0.125 x 15^{4}*15*^{2}= 225*15*^{4}= 225^{2}= (22x23) affix 25*=23 x (20 +2) = 460+46 affix 25 = 50625**15*^{5}= 50625 x (10 +5) = 506250 + 253125*= 75375*For the second term;*50625 x 5 = 253125**253125 x 0.125 = 253125x (0.1+ 0.2+0.005)**= 25312.5**5062.5*__1265.625__*31640.625*Thus;*15.125*^{5}= 739375__31640__**791015**The six digit value ofobtained with a scientific calculator is*15.125*^{5}. The error in the approximation is only 0.067% of the true value.*791547*For base numbers with one digit or more than two digit integer part, the number should be converted to a Special Exponential Notation form containing a two digit integer part of the significant figure before transforming it to the equivalent binomial form.**Examples:**1*. 2.374*^{5}=(23.74 x 10^{-1})^{5}=23.74^{5}x 10^{4}^{ }*=(23 + 0.74)*^{5}x 10^{4}^{ }2.*376*^{4}= (37.6 x 10)^{4}= 37.6^{4}x 10^{4}^{ }*= (37+ 0.6)*^{4}x 10^{4}In Example 1 the binomialmay be changed to*(23 + 0.74)*to improve the accuracy of the approximation since the ratio*(24 - 0.26)*is much smaller than*0.26/24*.*0.74/23*b.**INTERPOLATION**Interpolation may be used for evaluating exponentials. As in the previous method, the significant figure of the Exponential should have a two- digit integer part to maintain an acceptable accuracy.**Example:**

Evaluateby interpolation*26.25*^{3}**Solution:****X X**^{3}*26 17576***26.25 18102.75***27 19683*The value ofobtained with a scientific calculator is*26.25*^{3}. There is an error of*18087.89*in the approximation which is only 0.082*14.86**%*of the true value.^{}Your next algorithm is h`ow to raise a number containing a decimal part to an exponent that also has a decimal part.` - Dear Admin,Please remove my name from the group list.Best wishes for all the members of the group. may we found great mathematicians from this group.Best regardsFeroz
**From:**Diosdado |Fragata <dadofragata@...>**To:**"math_club@yahoogroups.com" <math_club@yahoogroups.com>**Sent:**Wednesday, November 2, 2011 9:28 AM**Subject:**[math_club] ALTERNATIVE ALGORITHM No.14HELLO!Here is your alternative algorithm for today:**ALTERNATIVE ALGORITHM # 14****RAISING A NUMBER TO AN INTEGER EXPONENT**The usual method of raising a number to an integer exponent is done by multiplying the number by itself successivelytimes as in*(n-1)**2*^{5}^{ }=x*2*x*2*x*2*x*2*=*2*. This method is manageable if the base number is not more than two digits. But if the base number is more than two digits, the task becomes exasperating. However, there are manual methods for approximating the values of exponentials.*32*a.**THE DIFFERENTIAL EXCESS METHOD**If a number is expressed as a binomial which is raised to exponent, the binomial expansion is:*n**(a*__+__b)^{n}= a^{n}__+__nba^{n-1}+ n(n-1)b^{2}a^{n-2}__+__…b^{n}Ifis very small compared to*b*, the approximate value of the binomial is equal to the sum of the values of the first two terms in the expansion thus:*a**(***a**__+__b)^{n}= a^{n}__+__n ba^{n-1}(5)The value obtained by using Eq.(5) is considered fairly accurate for as long asis at least a two digit integer and*a*is the decimal complement of the given base number.*b***Example:**Find the value of*15.125*^{5}**Solution:**

For the first term;*15.125*^{5}= (15 + 0.125)^{5}= 15^{5}+ 5 x 0.125 x 15^{4}*15*^{2}= 225*15*^{4}= 225^{2}= (22x23) affix 25*=23 x (20 +2) = 460+46 affix 25 = 50625**15*^{5}= 50625 x (10 +5) = 506250 + 253125*= 75375*For the second term;*50625 x 5 = 253125**253125 x 0.125 = 253125x (0.1+ 0.2+0.005)**= 25312.5**5062.5*__1265.625__*31640.625*Thus;*15.125*^{5}= 739375__31640__**791015**The six digit value ofobtained with a scientific calculator is*15.125*^{5}. The error in the approximation is only 0.067% of the true value.*791547*For base numbers with one digit or more than two digit integer part, the number should be converted to a Special Exponential Notation form containing a two digit integer part of the significant figure before transforming it to the equivalent binomial form.**Examples:**1*. 2.374*^{5}=(23.74 x 10^{-1})^{5}=23.74^{5}x 10^{4}^{ }*=(23 + 0.74)*^{5}x 10^{4}^{ }2.*376*^{4}= (37.6 x 10)^{4}= 37.6^{4}x 10^{4}^{ }*= (37+ 0.6)*^{4}x 10^{4}In Example 1 the binomialmay be changed to*(23 + 0.74)*to improve the accuracy of the approximation since the ratio*(24 - 0.26)*is much smaller than*0.26/24*.*0.74/23*b.**INTERPOLATION**Interpolation may be used for evaluating exponentials. As in the previous method, the significant figure of the Exponential should have a two- digit integer part to maintain an acceptable accuracy.**Example:**

Evaluateby interpolation*26.25*^{3}**Solution:****X X**^{3}*26 17576***26.25 18102.75***27 19683*The value ofobtained with a scientific calculator is*26.25*^{3}. There is an error of*18087.89*in the approximation which is only 0.082*14.86**%*of the true value.^{}Your next algorithm is h`ow to raise a number containing a decimal part to an exponent that also has a decimal part.`