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Home  >>  CBSE XI  >>  Math  >>  Binomial Theorem
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Using Binomial Theorem, evaluate $101^4$

$\begin{array}{1 1} 104060401 \\ 986453 \\ 357975 \\ 4790669 \end{array} $

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Toolbox:
  • Express the given number as the sum or difference of two numbers whose powers are easier to evaluate. Then use binomial theorem as follows:
  • $(a+b)^n = \large \sum \limits_{k=0}^{n}\; $$^n \large C$$_k \; a^{n-k}b^k =\; ^n \large C$$_0\;a^nb^0 +\; ^n \large C$$_1\;a^{n-1}b^1+.... ^n\large C$$_n\;a^0b^n$, where $b^0 = 1 = a^{n-n}$
$101 = 100 + 1 \rightarrow 101^4 = (100+1)^4$. We can now use Binomial Theorem to evaluate this.
$(100+1)^4 = ^4 \large C$$_0\;100^{4-1}1^0 +.^4 \large C$$_1\;100^{4-1}1^1 +.^4 \large C$$_2\;100^{4-2}1^2 +.^4 \large C$$_3\;100^{4-3}1^3 + ^4 \large C$$_4\;100^{4-4}1^4 $
$\qquad = 100^4 + 4(100^3) + 6(100^2) +4(100) + 1$
$\qquad = 100000000+4000000+60000+400+1$
$\qquad = 104060401$
answered Apr 10, 2014 by balaji.thirumalai
 

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