Can anyone explain the binomial expansion of terms raised to negative or fractional powers - Maths - Binomial Theorem

Question

Can anyone explain the binomial expansion of terms raised to
negative or fractional powers?

Chetna Khanna · Accepted Answer

Dear Student

The binomial expansion for negative integer power, –n is:

$(x + a)^{- n} = Σ_{K = 0}^{\infty} (- 1)^{K} (\begin{matrix} n + K–1 \\ K \end{matrix}) x^{K} a^{- n - K}$

Example:

Expansion of (1 + x)^–3 is as follows:

$(1 + x)^{- 3} = 1 - 3 x + \frac{1}{2} \times 3 \times (3 + 1) x^{2} - \frac{1}{6} \times 3 \times (3 + 1) \times (3 + 2) x^{3} = 1 - 3 x + 6 x^{2} - 10 x^{3}$

The binomial expansions for fractional power is same as for integers.

$(a + b)^{n} = Σ_{K = 0}^{n}^{n} C_{K} a^{n - k} b^{k}$

Example:

Expansion of $(1 + x)^{\frac{3}{2}}$ is as follows:

${(1 + x)}^{\frac{3}{2}} = 1 + \frac{3}{2} x + \frac{\frac{3}{2} \times \frac{1}{2}}{2!} x^{2} + \frac{\frac{3}{2} \times \frac{1}{2} \times (\frac{- 1}{2})}{3!} x^{3} + ......... = 1 + \frac{3}{2} x + \frac{3}{8} x^{2} - \frac{1}{16} x^{3} + .................$

$[Note: (1 + x)^{n} = 1 + n x + \frac{n (n - 1)}{2} x^{2} + \frac{n (n - 1) (n - 2)}{3!} x^{3} + .............]$

Cheers!!

Can anyone explain the binomial expansion of terms raised to negative or fractional powers?