The force of gravity on a mass $m$ is $F=-GMm/x^2$. With $G=6\cdot10^{-17}$ and Earth mass $M=6\cdot10^{24}$ and rocket mass $m=1000$, compute the work to lift the rocket from $x=6400$ to $x=6500$. (The units are kgs and kms and Newtons, giving work in Newton-kms).
The force required to lift the rocket is $F=\frac{GMm}{x^2}=\frac{36\cdot10^{10}}{x^2}$. The work is $\int_{6400}^{6500}\frac{36\cdot10^{10}}{x^2}\;dx=\left.-\frac{36\cdot10^{10}}x\right|_{6400}^{6500}=36\cdot10^{10}\left(\frac1{6400}-\frac1{6500}\right)=\frac9{104}\cdot10^7\approx865384.6154$.
However, the answer key says that it is 864,000 Nkm. Why is there a difference of 1384.6154?
Just a guess, but $$\int_{6400}^{6500}\frac{1}{x^2}\,dx \approx 2.4038\times10^{-6}.$$ If you round that off to $2.40 \times 10^{-6}$ and then multiply by $3.6 \times 10^{11},$ you get $8.64 \times 10^5.$
I would question how you're supposed to decide to round to three significant digits when two of the inputs to the problem are stated with only one significant digit. But given those very rough approximations in the input, a discrepancy of less than $0.2\%$ such as you found between your answer and the answer key is not significant.