Expanding \((a+bx)^n\)
\(\sqrt{ 4+x }\)
- \((4+x)^{\frac{1}{2}}\)
- \(4^{\frac{1}{2}}\left( 1+\frac{x}{4} \right)^{\frac{1}{2}}\)
- \(2\left( 1+\frac{x}{4} \right)^{\frac{1}{2}}\)
- We can now expand this using the regular binomial expansion.
- \(1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3\dots\)
- Multiply expansion by 2
\(\frac{1}{(2+3x)^2}\)
- \((2+3x)^{-2}\)
- \(2^{-2}\left( 1+\frac{3}{2}x \right)^{-2}\)
- \(\frac{1}{4}\left( 1+\frac{3}{2} \right)^{-2}\)
- Expand using regular binomial expansion: \(1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3\dots\)
- Multiple expansion by \(\frac{1}{4}\)