Expanding (i) and Solving (ii): Find the Solution

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chwala
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Homework Statement
See attached problem with ms
Relevant Equations
binomial theorem
Find question here and ms... In part ##i## we could just as well expand directly hence reason why i am sharing...

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My direct expansion for part (i),

$$(4-x)^{-\frac{1}{2}} =4^{-\frac{1}{2}}+\frac{(\frac{-1}{2}⋅4^{-\frac{3}{2}}⋅-x)}{1!}+\frac {(\frac{3}{2}⋅\frac{1}{2}⋅4^-\frac{5}{2}⋅(-x)^2)}{2!}=\frac{1}{2}+\frac{1}{16}x+\frac{3}{256}x^2+...$$

part (ii) follows directly from (i),

##(a+bx)(\frac{1}{2}+\frac{1}{16}x+\frac{3}{256}x^2+...)=16-x...##
##\frac{1}{2}a+\frac{1}{16}ax+\frac{3}{256}ax^2+\frac{1}{2}bx+\frac{1}{16}bx^2+\frac{3}{256}bx^3+...=16-x...##

giving us the two simultaneous equations indicated. cheers
 
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As an alternative way
(\frac{a+bx}{\sqrt{4-x}})_{x=0}=\frac{a}{2}=16
a=32
(\frac{32+bx}{\sqrt{4-x}})'_{x=0}=\frac{b+4}{2}=-1
b=-6
 
anuttarasammyak said:
As an alternative way
(\frac{a+bx}{\sqrt{4-x}})_{x=0}=\frac{a}{2}=16
a=32
(\frac{32+bx}{\sqrt{4-x}})'_{x=0}=\frac{b+4}{2}=-1
b=-6
Smart move there @anuttarasammyak ...just wondering if this approach would be acceptable, i.e substituting ##x=0## directly ... are we not supposed to make use of part (i) though?

I can see that its a B mark, thus acceptable...cheers
 
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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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