Step by step process on how to solve this problem

Step-1:  Remember or look up the formula for the force of gravity between two objects.    F = G · m₁ · m₂ / R²

Step-2:  Remember or look up the value of G. G = 6.67 x 10⁻¹¹ m³·kg/s²

Step-3: Write the numbers you know into the formula.

(1.989 x 10²⁰ Newtons) =

(6.67 x 10⁻¹¹ mtr³·kg/s²) · (5.9742 x 10²⁴ kg) · (moon mass) / (3.84 x 10⁸ mtr)²

Step-4:  Sit back, relax, take your time, look this mess over, carefully, end-to-end, and decide how to solve it for (moon mass) .

Step-5: Divide each side by (6.67x10⁻¹¹mtr³·kg/s²)·(5.9742x10²⁴kg)/(3.84x10⁸mtr)²:

Moon mass =

(1.989x10²⁰ Newtons)·(3.84x10⁸ mtr)²/(6.67x10⁻¹¹ mtr³·kg/s²)·(5.9742x10²⁴ kg)

Step-6: Crunch the numbers.  Be careful to KEEP all the units as you go along. When you're done, the units of your answer will be the first instant indication if you made a mistake.  You're looking for the MASS of the moon.  If the answer doesn't have units of 'kg', then that'll be an immediate red flag, telling you that there's been a mistake somewhere.

Moon mass =

(1.989x10²⁰ Newtons)·(3.84x10⁸ mtr)²/(6.67x10⁻¹¹ mtr³·kg/s²)·(5.9742x10²⁴ kg)

Collect the numbers, and collect the units:

Moon mass = (1.989x10²⁰ · (3.84x10⁸)² / (6.67x10⁻¹¹ · 5.9742x10²⁴)

(kg-mtr/s² · mtr²)/(mtr³·kg/s² · kg)

Moon mass = (1.989 · 3.84² x 10³⁶) / (6.67 · 5.9742 x 10¹³)

(kg-mtr/s² · mtr²)/(mtr³·kg/s² · kg)

Moon mass = 0.736 x 10²³ kg

Moon mass = 7.36 x 10²² kg

Step 7: Look it up in a book or online.  See if you're anywhere close.

When I search "moon mass" on Floogle, the first hit says

" 7.348 x 10²² kg " .

yay !  I'm satisfied.

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