Question

Use a binomial theorem to find the value of ${(102)^3}$.

Answer 1

Hint – In order to solve this problem you need to write 102 = 100+2. Then apply the binomial theorem to get the value of ${(102)^3}$.

Complete step-by-step answer:

We have to find the value of ${(102)^3}$ using the binomial theorem.

We can write ${(102)^3}$ as ${(100 + 2)^3}$.

We know that ${(a + b)^n} = {\,^n}{C_0}{a^n}{b^0} + {\,^n}{C_1}{a^{n - 1}}{b^1} + ....... + {\,^n}{C_n}{a^0}{b^n}$. (Binomial Expansion)

So we can apply the same expansion in ${(100 + 2)^3}$.

So, $
  {(100 + 2)^3} = {\,^3}{C_0}{100^3}{2^0} + {\,^3}{C_1}{100^{^{3 - 1}}}{2^1} + {\,^3}{C_2}{100^{3 - 2}}{2^2} + {\,^3}{C_3}{100^{3 - 3}}{2^3} \\
  {(100 + 2)^3} = \dfrac{{3!}}{{0!(3 - 0)!}}(1000000) + \dfrac{{3!}}{{1!(3 - 1)!}}(20000) + \dfrac{{3!}}{{2!(3 - 2)!}}(400) + \dfrac{{3!}}{{3!(3 - 3)!}}(8) \\
  {(100 + 2)^3} = 1000000 + 60000 + 1200 + 8 \\
  {(100 + 2)^3} = 1061208. \\
$
Hence, the answer to this question is 1061208.

Note – When you have asked to expand something with the help of binomial expansion then break it into two parts to apply the binomial expansion and then use the expansion ${(a + b)^n} = {\,^n}{C_0}{a^n}{b^0} + {\,^n}{C_1}{a^{n - 1}}{b^1} + ....... + {\,^n}{C_n}{a^0}{b^n}$ and solve to get the answer to this type of question.