Suppose we have a cube of `1.00 cm` length. It is cut in all three directions, so as to produce eight cubes, each `0.50 cm` on edge length. Then suppose these `0.50` cm cubes are each subdivided into eight cubes `0.25` cm on edge length, and so on. How many of these successive subdivisions are required before the cubes are reduced in size to colloidal dimensions of `100 nm`.

Answer

To solve the problem step by step, we will analyze how the size of the cubes changes with each subdivision and determine how many subdivisions are needed to reach a size of 100 nm.

### Step 1: Understand the Initial Size
The initial size of the cube is given as 1.00 cm.

### Step 2: Determine the Size After Each Subdivision
When the cube is cut into 8 smaller cubes, each edge length is halved.
- After the first subdivision:
\[ \text{Size} = \frac{1.00 \, \text{cm}}{2} = 0.50 \, \text{cm} \]
- After the second subdivision:
\[ \text{Size} = \frac{0.50 \, \text{cm}}{2} = 0.25 \, \text{cm} \]
- After the third subdivision:
\[ \text{Size} = \frac{0.25 \, \text{cm}}{2} = 0.125 \, \text{cm} \]

### Step 3: Generalize the Size After n Subdivisions
We can express the size of the cube after n subdivisions as:
\[ \text{Size after n subdivisions} = \frac{1.00 \, \text{cm}}{2^n} = \frac{1}{2^n} \, \text{cm} \]

### Step 4: Convert Colloidal Size to Centimeters
The colloidal dimension is given as 100 nm. We need to convert this to centimeters:
\[ 100 \, \text{nm} = 100 \times 10^{-9} \, \text{m} = 100 \times 10^{-7} \, \text{cm} = 1 \times 10^{-5} \, \text{cm} \]

### Step 5: Set Up the Equation
We want to find n such that:
\[ \frac{1}{2^n} \, \text{cm} = 1 \times 10^{-5} \, \text{cm} \]
This simplifies to:
\[ 2^n = 10^5 \]

### Step 6: Take Logarithms
Taking the logarithm on both sides:
\[ n \log(2) = 5 \log(10) \]
Since \(\log(10) = 1\):
\[ n \log(2) = 5 \]

### Step 7: Solve for n
Now, we can solve for n:
\[ n = \frac{5}{\log(2)} \]
Using \(\log(2) \approx 0.301\):
\[ n \approx \frac{5}{0.301} \approx 16.61 \]

### Step 8: Round to the Nearest Whole Number
Since n must be a whole number, we round 16.61 to 17.

### Conclusion
Thus, the number of subdivisions required to reduce the cubes to colloidal dimensions of 100 nm is approximately **17**.

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