Calculate the distance from the earth to the point where the gravitational field due to the earth and the moon cancel out. Given that the earth-moon distance is `3.8 xx 10^(8) m` and the mass of earth is 81 times that of moon.

Explanation

To solve the problem of finding the distance from the Earth to the point where the gravitational fields due to the Earth and the Moon cancel each other out, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables:**
- Let \( M_m \) be the mass of the Moon.
- Then the mass of the Earth \( M_e = 81 M_m \).
- The distance between the Earth and the Moon is given as \( d = 3.8 \times 10^8 \) m.

2. **Set Up the Problem:**
- Let \( x \) be the distance from the Earth to the point where the gravitational fields cancel out.
- The distance from the Moon to this point will then be \( d - x = 3.8 \times 10^8 - x \).

3. **Write the Gravitational Field Equations:**
- The gravitational field due to the Earth at distance \( x \) is given by:
\[ E_e = \frac{G M_e}{x^2} = \frac{G (81 M_m)}{x^2} \]
- The gravitational field due to the Moon at distance \( d - x \) is given by:
\[ E_m = \frac{G M_m}{(d - x)^2} \]

4. **Set the Gravitational Fields Equal:**
- For the fields to cancel out, we set \( E_e = E_m \):
\[ \frac{G (81 M_m)}{x^2} = \frac{G M_m}{(3.8 \times 10^8 - x)^2} \]
- We can cancel \( G \) and \( M_m \) from both sides:
\[ \frac{81}{x^2} = \frac{1}{(3.8 \times 10^8 - x)^2} \]

5. **Cross Multiply:**
- Cross multiplying gives:
\[ 81 (3.8 \times 10^8 - x)^2 = x^2 \]

6. **Expand and Rearrange:**
- Expanding the left side:
\[ 81 (3.8^2 \times 10^{16} - 7.6 \times 10^8 x + x^2) = x^2 \]
- This simplifies to:
\[ 81 \times 14.44 \times 10^{16} - 81 \times 7.6 \times 10^8 x + 81 x^2 = x^2 \]
- Rearranging gives:
\[ 80 x^2 - 81 \times 7.6 \times 10^8 x + 81 \times 14.44 \times 10^{16} = 0 \]

7. **Solve the Quadratic Equation:**
- Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 80 \), \( b = -81 \times 7.6 \times 10^8 \), and \( c = 81 \times 14.44 \times 10^{16} \).
- Calculate \( b^2 - 4ac \) and then find \( x \).

8. **Calculate the Values:**
- After performing the calculations, we find:
\[ x \approx 3.42 \times 10^8 \text{ m} \]

### Final Answer:
The distance from the Earth to the point where the gravitational fields due to the Earth and the Moon cancel out is approximately \( 3.42 \times 10^8 \) meters.