Question 825831
1) I am not sure exactly what you are expected to do to evaluate {{{root(4,624)}}} without using a calculator, using binomial theorem, but this is what I would do.
{{{4^4=256}}} and {{{5^4=625}}}
Since 624 is so close to 625, i would expect {{{root(4,624)}}} to be very close to 5, so let's try 4.9.
{{{4.9^4=(5-0.1)^4=g^4-4*5^3*0.1+6*5^2*0.1^2-4*5*0.1^3+0.1^4=625-4*125*0.1+6*25*0.01-4*5*.001+.0001=625-50+1.5-.02+.0001=576.4801}}}
Since 624 is much closer to 625 than to 576.48, I would expect {{{root(4,624)}}} to be much closer to 5 than to 4.9, so I would try 4.99.
{{{4.99^4=(5-0.01)^4=g^4-4*5^3*0.01+6*5^2*0.01^2-4*5*0.01^3+0.01^4=625-4*125*0.01+6*25*0.0001-4*5*0.000001+0.00000001}}}
= about{{{625-5+0.015=620.015}}}
At this point, knowing that {{{4.99^4=about620.015}}} and {{{5.00^4=625.000}}}, I would estimate {{{root(4,624)=4.998}}} by assuming that {{{f(x)=root(4,x)}}} is approximately linear between 4.990 and 5.000.
 
2) There is nothing that can be evaluated in
{{{1/(1+x)^1/2=1/sqrt(1+x)}}} , since we do not have a value for {{{x}}} .
That expression can be rationalized as
{{{1/(1+x)^1/2=1/sqrt(1+x)=(1/sqrt(1+x))*(sqrt(1+x)/sqrt(1+x))=1*sqrt(1+x)/(sqrt(1+x)*sqrt(1+x))=sqrt(1+x)/(sqrt(1+x))^2=sqrt(1+x)/(1+x)}}}