Question

Using Binomial Theorem, evaluate \((102)^5\).

Answer 1

\(102\) can be expressed as the sum or difference of two numbers whose powers are easier to calculate and then, binomial theorem can be applied.
It can be written that, \(102 = 100 +2\)
∴ \((102)^5=(100+2)^5\)
= \(^5C_0(100)^5+ ^5C_1(100)^4 (2) + ^5C_2(100)^3 (2)^2 + ^5C_3(100)^2(2)^3 + ^5C_4(100) (2)^4+ ^5C_5(2)^5\)
=\((100)^5+5(100)^4(2)+10(100)^3 (2)^2+10(100)^2(2)^3 +5(100)(2)^4+(2)^5\)
=\(10000000000+1000000000+40000000+800000+8000+32\)
=\(11040808032\)