Solution:

The weight of an object on the Moon is given by:

\[
W_{\text{moon}} = \frac{1}{6} W_{\text{earth}}
\]

Weight is related to the gravitational acceleration \( g \) by:

\[
W = mg
\]

Thus,

\[
g_{\text{moon}} = \frac{1}{6} g_{\text{earth}}
\]

The formula for gravitational acceleration is:

\[
g = \frac{GM}{R^2}
\]

Where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Moon,
- \( R \) is the radius of the Moon.

Given:
- \( g_{\text{earth}} = 9.8 \, \text{m/s}^2 \),
- \( g_{\text{moon}} = \frac{1}{6} \times 9.8 = 1.633 \, \text{m/s}^2 \),
- \( R_{\text{moon}} = 1.768 \times 10^6 \, \text{m} \).

Now, solve for \( M_{\text{moon}} \) using:

\[
g_{\text{moon}} = \frac{G M_{\text{moon}}}{R_{\text{moon}}^2}
\]

Rearranging for \( M_{\text{moon}} \):

\[
M_{\text{moon}} = \frac{g_{\text{moon}} R_{\text{moon}}^2}{G}
\]

Substitute values:

\[
M_{\text{moon}} = \frac{1.633 \times (1.768 \times 10^6)^2}{6.674 \times 10^{-11}}
\]

Calculating this gives:

\[
M_{\text{moon}} \approx 7.35 \times 10^{22} \, \text{kg}
\]

Thus, the mass of the Moon is approximately \( 7.35 \times 10^{22} \, \text{kg} \).