Find cube root of $63$ to three decimal places 
Find $\sqrt[3]{63}$ to three decimal places (without calculus or a calculator). 

I am unable to find it. How to get the result by hand calculation? Please help.

$64^{1/3} = 4$, so the answer is a bit less than $4$.
$(4 - x)^3 = 64 - 48 x + 12 x^2 - x^3$.  Looking just at the first two terms, we want $x \approx 1/48 \approx 0.021$, i.e. the cube root is approximately $3.979$.  This does turn out to be accurate to three decimal places.
