The problem is asking for the mass of a planet based on its radius and surface gravity. To solve this, we use the formula for surface gravity:

$g = \frac{GM}{R^2}$

Where:

• $g$ is the surface gravity ($6.40 \, \text{m/s}^2$),
• $G$ is the gravitational constant ($6.674 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}$),
• $M$ is the mass of the planet (which we need to find),
• $R$ is the radius of the planet ($1.25 \times 10^6 \, \text{m}$).

Rearranging the formula to solve for the mass $M$:

$M = \frac{gR^2}{G}$

Step-by-Step Calculation:

1. Plug in the given values:

   • $g = 6.40 \, \text{m/s}^2$,
   • $R = 1.25 \times 10^6 \, \text{m}$,
   • $G = 6.674 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}$.

2. Calculate the mass: $M = \frac{(6.40) \times (1.25 \times 10^6)^2}{6.674 \times 10^{-11}}$

Let me calculate that for you.The mass of the planet is approximately $1.50 \times 10^{23} \, \text{kg}$.

So, the coefficient (green) is 1.50, and the exponent (yellow) is 23.

Would you like further details or have any questions about this solution?

Here are 5 related questions:

1. How does the surface gravity of a planet relate to its mass and radius?
2. What would happen to the mass if the surface gravity doubled while keeping the radius constant?
3. How can we calculate the surface gravity of a planet if the mass is known?
4. What is the significance of the gravitational constant $G$ in planetary calculations?
5. How does the radius of a planet affect its surface gravity?

Tip: When solving gravitational problems, keep track of units to ensure correct results, especially when using scientific notation.