Find the distance between the origin and `(3sqrt5,-2)`

Answer

To find the distance between the origin (0, 0) and the point (3√5, -2), we can use the distance formula. The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

### Step-by-Step Solution:

1. **Identify the coordinates**:
- The origin \(O\) has coordinates \((0, 0)\).
- The point \(A\) has coordinates \((3\sqrt{5}, -2)\).

2. **Assign values to the variables**:
- Let \(x_1 = 0\), \(y_1 = 0\) (coordinates of the origin).
- Let \(x_2 = 3\sqrt{5}\), \(y_2 = -2\) (coordinates of point A).

3. **Substitute the values into the distance formula**:
\[ d = \sqrt{(3\sqrt{5} - 0)^2 + (-2 - 0)^2} \]

4. **Simplify the expression**:
\[ d = \sqrt{(3\sqrt{5})^2 + (-2)^2} \]

5. **Calculate the squares**:
- \((3\sqrt{5})^2 = 3^2 \times (\sqrt{5})^2 = 9 \times 5 = 45\)
- \((-2)^2 = 4\)

6. **Add the results**:
\[ d = \sqrt{45 + 4} = \sqrt{49} \]

7. **Find the square root**:
\[ d = 7 \]

### Final Answer:
The distance between the origin and the point \((3\sqrt{5}, -2)\) is \(7\).