1. **State the problem:** Calculate the velocity $v$ given by the formula $$v=\sqrt{\frac{6.67\times10^{-11} \times 5.97\times10^{24}}{400000 + 6.37\times10^{6}}}$$ where the numerator is the product of the gravitational constant and Earth's mass, and the denominator is the sum of 400000 meters and Earth's radius in meters. 2. **Formula and explanation:** This formula resembles the orbital velocity formula $$v=\sqrt{\frac{GM}{r}}$$ where $G$ is the gravitational constant, $M$ is the mass of the Earth, and $r$ is the distance from the Earth's center. 3. **Calculate the denominator:** $$400000 + 6.37\times10^{6} = 400000 + 6370000 = 6770000$$ meters. 4. **Calculate the numerator:** $$6.67\times10^{-11} \times 5.97\times10^{24} = (6.67 \times 5.97) \times 10^{-11 + 24} = 39.8199 \times 10^{13} = 3.98199 \times 10^{14}$$ 5. **Calculate the fraction inside the square root:** $$\frac{3.98199 \times 10^{14}}{6.77 \times 10^{6}} = \frac{3.98199}{6.77} \times 10^{14 - 6} = 0.5883 \times 10^{8} = 5.883 \times 10^{7}$$ 6. **Calculate the square root:** $$v = \sqrt{5.883 \times 10^{7}} = \sqrt{5.883} \times \sqrt{10^{7}} \approx 2.425 \times 10^{3.5} = 2.425 \times 10^{3} \times 10^{0.5} = 2.425 \times 1000 \times 3.162 = 2.425 \times 3162 = 7667.35$$ meters per second. **Final answer:** $$v \approx 7667.35$$ meters per second.