Find the distance from the origin to each of the point:
`(0, 4,-4)`

Answer

To find the distance from the origin to the point (0, 4, -4), we can use the distance formula in three-dimensional space. The distance \( d \) from the origin (0, 0, 0) to a point \( (x, y, z) \) is given by the formula:

\[ d = \sqrt{(x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2} \]

Where \( (x_1, y_1, z_1) \) are the coordinates of the origin, which are (0, 0, 0) in this case.

### Step 1: Identify the coordinates
The coordinates of the point are \( (0, 4, -4) \) and the coordinates of the origin are \( (0, 0, 0) \).

### Step 2: Substitute the coordinates into the distance formula
Substituting the values into the distance formula, we have:

\[ d = \sqrt{(0 - 0)^2 + (4 - 0)^2 + (-4 - 0)^2} \]

### Step 3: Simplify the expression
Now, simplifying the expression:

\[ d = \sqrt{0^2 + 4^2 + (-4)^2} \]
\[ d = \sqrt{0 + 16 + 16} \]
\[ d = \sqrt{32} \]

### Step 4: Simplify \( \sqrt{32} \)
Next, we simplify \( \sqrt{32} \):

\[ \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \]

### Conclusion
Thus, the distance from the origin to the point \( (0, 4, -4) \) is:

\[ d = 4\sqrt{2} \]

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