maths Data management question?

Question

Find the vaule of (21)^(1/4) using the Binomial Theorem correct to 3 decimal places.


I don't understand what it is asking ?

Answer 1

The binomial theorem says

(a+b)^p = a^p + pa^(p-1)*b + p(p-1)a^(p-2)*(b^2)/2! + ... for a>b

For a fractional power, this is of use for a problem like yours if we can evaluate a^p exactly. So you want a number near 21 for which we know the 4th root exactly. There is no single correct value, but most likely you're expected to use 16, whose 4th root is 2. In short, use the binomial theorem with a=16, b=5, and p=1/4:

21^(1/4) = 16^1/4 + (1/4)16^(-3/4)*5 + (1/4)(-3/4)16^(-7/4)(5^2)/2! + ...

or equivalently

21^(1/4) = (16+5)^(1/4) = (16)^(1/4) (1 + 5/16)^(1/4) = 2[1 + (1/4)(5/16) + (1/4)(-3/4)(5/16)^2 / 2! + ...

After the first term, you get an alternating series with terms decreasing in magnitude. We know that the magnitude of the error in truncating after N terms is less than the magnitude of the (N+1)st term, so you can stop as soon as the magnitude of a term is less than or equal to 5*10^(-4), which assures three-decimal accuracy.