Find the real cube root of
`38sqrt(14)-100sqrt(2)`.

Answer

To find the real cube root of the expression \( 38\sqrt{14} - 100\sqrt{2} \), we will follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ 38\sqrt{14} - 100\sqrt{2} \]

### Step 2: Factor Out Common Terms
We can factor out \( -2\sqrt{2} \) from the expression:
\[ = -2\sqrt{2} \left( 19\sqrt{7} - 50 \right) \]

### Step 3: Identify the Cube Root
We need to express this in a form suitable for taking the cube root. We can write:
\[ \sqrt[3]{38\sqrt{14} - 100\sqrt{2}} = \sqrt[3]{-2\sqrt{2} \left( 19\sqrt{7} - 50 \right)} \]

### Step 4: Set Up for Cube Root Calculation
Let:
\[ x = \sqrt[3]{19\sqrt{7} - 50} \]
Then we can express our cube root as:
\[ \sqrt[3]{-2\sqrt{2}} \cdot x \]

### Step 5: Calculate the Cube Root of \(-2\sqrt{2}\)
The cube root of \(-2\sqrt{2}\) can be calculated as:
\[ \sqrt[3]{-2\sqrt{2}} = -\sqrt[3]{2} \cdot \sqrt[3]{\sqrt{2}} = -\sqrt[3]{2} \cdot 2^{1/6} \]

### Step 6: Find the Value of \(x\)
To find \(x\), we need to compute:
\[ 19\sqrt{7} - 50 \]
Since \(19\sqrt{7} \approx 50.3\), we find that:
\[ 19\sqrt{7} - 50 \approx 0.3 \]
Thus, we can approximate:
\[ x \approx \sqrt[3]{0.3} \]

### Step 7: Combine the Results
Finally, we combine our results:
\[ \sqrt[3]{38\sqrt{14} - 100\sqrt{2}} \approx -\sqrt[3]{2} \cdot 2^{1/6} \cdot \sqrt[3]{0.3} \]

### Final Result
The real cube root of \( 38\sqrt{14} - 100\sqrt{2} \) is:
\[ -\sqrt[3]{2} \cdot 2^{1/6} \cdot \sqrt[3]{0.3} \]