A 25 kg instrument is lifted from the ground to 250 km up…

Step 1. Exact. $25\ \text{kg} \times 9.8\ \text{m/s}^2 \times 6.4 \times 10^{6}\ \text{m} \times \dfrac{250\ \text{km}}{6650\ \text{km}} = 5.89 \times 10^{7}\ \text{J}$.

Step 2. Shortcut. $mgh = 25\ \text{kg} \times 9.8\ \text{m/s}^2 \times 2.5 \times 10^{5}\ \text{m} = 6.13 \times 10^{7}\ \text{J}$.

Step 3. Error. $\dfrac{mgh}{\text{exact}} = \dfrac{R + h}{R} = 1.039$: $3.9\%$ too big.