What is the distance between `(-2, -4)` and `(3, 8)`?

Answer

To find the distance between the points \((-2, -4)\) and \((3, 8)\), we will use the distance formula. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step-by-Step Solution:

1. **Identify the coordinates**:
- Let \((x_1, y_1) = (-2, -4)\)
- Let \((x_2, y_2) = (3, 8)\)

2. **Substitute the coordinates into the distance formula**:
\[ d = \sqrt{(3 - (-2))^2 + (8 - (-4))^2} \]

3. **Simplify the expressions inside the parentheses**:
- Calculate \(3 - (-2)\):
\[ 3 + 2 = 5 \]
- Calculate \(8 - (-4)\):
\[ 8 + 4 = 12 \]

4. **Plug these values back into the formula**:
\[ d = \sqrt{(5)^2 + (12)^2} \]

5. **Calculate the squares**:
- \(5^2 = 25\)
- \(12^2 = 144\)

6. **Add the squares**:
\[ 25 + 144 = 169 \]

7. **Take the square root**:
\[ d = \sqrt{169} = 13 \]

### Final Answer:
The distance between the points \((-2, -4)\) and \((3, 8)\) is \(13\) units.

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