To find the sum of distances from the origin to the given points, we need to calculate the individual distances first and then add them up. The origin in a 3D coordinate system is the point \((0, 0, 0)\).

Distance Formula in 3D Space

The distance between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in three-dimensional space is given by the distance formula:

\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

Since we are calculating the distance from the origin \((0, 0, 0)\) to a point \((x, y, z)\), the formula simplifies to:

\[ D = \sqrt{(x - 0)^2 + (y - 0)^2 + (z - 0)^2} = \sqrt{x^2 + y^2 + z^2} \]

Calculating Distance to First Point: (0, 5, 5)

Let's find the distance from the origin \((0, 0, 0)\) to the first point \((0, 5, 5)\).

Here, \(x = 0, y = 5, z = 5\).

Using the simplified distance formula:

\[ D_1 = \sqrt{0^2 + 5^2 + 5^2} \]

\[ D_1 = \sqrt{0 + 25 + 25} \]

\[ D_1 = \sqrt{50} \]

To simplify \(\sqrt{50}\), we can factor out the largest perfect square, which is \(25\):

\[ D_1 = \sqrt{25 \times 2} \]

\[ D_1 = 5\sqrt{2} \]

Calculating Distance to Second Point: (5, 8, 6)

Next, we find the distance from the origin \((0, 0, 0)\) to the second point \((5, 8, 6)\).

Here, \(x = 5, y = 8, z = 6\).

Using the simplified distance formula:

\[ D_2 = \sqrt{5^2 + 8^2 + 6^2} \]

\[ D_2 = \sqrt{25 + 64 + 36} \]

\[ D_2 = \sqrt{125} \]

To simplify \(\sqrt{125}\), we can factor out the largest perfect square, which is \(25\):

\[ D_2 = \sqrt{25 \times 5} \]

\[ D_2 = 5\sqrt{5} \]

Summing the Distances

The question asks for the sum of distances from the origin to both points. So, we add the two calculated distances, \(D_1\) and \(D_2\):

\[ \text{Sum} = D_1 + D_2 \]

\[ \text{Sum} = 5\sqrt{2} + 5\sqrt{5} \]

We can factor out the common term \(5\) from both terms:

\[ \text{Sum} = 5(\sqrt{2} + \sqrt{5}) \]

Therefore, the sum of distances from the origin to (0, 5, 5) and (5, 8, 6) is \(5(\sqrt{2} + \sqrt{5})\).