1.5: Binomial approximation of square roots

(a) Expand $\displaystyle \sqrt{1-x}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$. Give each coefficient as an exact fraction in its lowest terms.

(b) Using your expansion with a suitable value of $\displaystyle x$, obtain an approximation, to 6 decimal places, of $\displaystyle \sqrt{0.92}$.

(c) Hence find an approximation, to 5 decimal places, of $\displaystyle \sqrt{23}$.