How do you use the Binomial Theorem to find the value of #99^4#?

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Answer 1 · 2018-05-03T10:05:14

#99^4=96059601#

We know that,

#color(blue)((x+a)^n=nC_0x^n+nC_1x^(n-1)a+nC_2x^(n- 2)a^2+...+nC_na^n#

Now,

#99^4=(100-1)^4=(100+(-1))^4#,

Take , #color(blue)(x=100 , a=-1 and n=4#

#99^4=4C_0(100)^4+4C_1(100)^3(-1)+4C_2(100)^2(-1)^2+4C_ 3(100)^1(-1)^3+4C_4 (-1)^4#

Here,

#4C_0=1,4C_1=4,4C_2=(4xx3)/(2xx1)=6,4C_3= (4xx3xx2)/(3xx2xx1)=4 ,4C_4=1and 100=10^2#

#=>99^4=1(10^8)-4(10^6)+6(10^4)-4(10^2)+1#

#=>99^4=100000000-4000000+60000-400+1#

#=>99^4=100060001-4000400#

#=>99^4=96059601#