The best way to learn binomial expansion is to try it out yourself! Let's use an example of expanding \( (x-3)^{1/2} \) to the 4th term.
Step 1: Rewriting the equation into the form \( (1+ax)^n \)
\( (x-3)^{1/2} = (3+x)^{1/2} = 3^{1/2} (1 + x/3)^{1/2} = \sqrt{3} (1 + x/3)^{1/2} \)
Step 2: Expanding using the formula \( 1 + nax + \frac{(n(n-1))}{2!}(ax)^2 + \frac{(n(n-1)(n-2))}{3!}(ax)^3 + \dots \)
$$\sqrt{3} \left(1 + \frac{1}{2}\left(\frac{x}{3}\right) + \frac{(1/2)(-1/2)}{2!} \left(\frac{x}{3}\right)^2 + \frac{(1/2)(-1/2)(-3/2)}{3!} \left(\frac{x}{3}\right)^3\right)$$
\( = \sqrt{3} \left(1 + \frac{x}{6} - \frac{x^2}{72} + \frac{x^3}{432}\right) \)
\( = \sqrt{3} + \frac{\sqrt{3}}{6} x - \frac{\sqrt{3}}{72} x^2 + \frac{\sqrt{3}}{432} x^3 \)
Finding the range of values of \( x \) where the expansion is valid
We must convert our equation into the form \( (1+ax)^n \), which we have done earlier
\( |ax| < 1 \)
\( |\frac{x}{3}| < 1 \)
\( |x| < 3 \)
Therefore, \( -3 < x < 3 \)
Pro tip: You can check if your expansion is correct by picking a number within the valid range and substituting it into the original equation and your expanded equation. If the results you get are similar, you've done the expansion correctly.