Simplify:
`(1^(2) + 2^(2) + 3^(2) + 4^(2))^(1//3) xx (1^(3) + 2^(3) + 3^(3) + 4^(3))^(1//3) xx (2^(2) + 3^(2) - 5 xx 2)^(2//3)`

Options

A. 52
B. 20
C. 13
D. 30

Answer

To simplify the expression `(1^(2) + 2^(2) + 3^(2) + 4^(2))^(1/3) × (1^(3) + 2^(3) + 3^(3) + 4^(3))^(1/3) × (2^(2) + 3^(2) - 5 × 2)^(2/3)`, we will follow these steps:

### Step 1: Calculate \(1^2 + 2^2 + 3^2 + 4^2\)
\[ 1^2 = 1, \quad 2^2 = 4, \quad 3^2 = 9, \quad 4^2 = 16 \]
Now, add them together:
\[ 1 + 4 + 9 + 16 = 30 \]
So, \(1^2 + 2^2 + 3^2 + 4^2 = 30\).

### Step 2: Raise the result to the power of \(1/3\)
\[ (30)^{1/3} \]

### Step 3: Calculate \(1^3 + 2^3 + 3^3 + 4^3\)
\[ 1^3 = 1, \quad 2^3 = 8, \quad 3^3 = 27, \quad 4^3 = 64 \]
Now, add them together:
\[ 1 + 8 + 27 + 64 = 100 \]
So, \(1^3 + 2^3 + 3^3 + 4^3 = 100\).

### Step 4: Raise the result to the power of \(1/3\)
\[ (100)^{1/3} \]

### Step 5: Calculate \(2^2 + 3^2 - 5 \times 2\)
\[ 2^2 = 4, \quad 3^2 = 9, \quad 5 \times 2 = 10 \]
Now, substitute and simplify:
\[ 4 + 9 - 10 = 3 \]
So, \(2^2 + 3^2 - 5 \times 2 = 3\).

### Step 6: Raise the result to the power of \(2/3\)
\[ (3)^{2/3} \]

### Step 7: Combine all parts together
Now we have:
\[ (30)^{1/3} \times (100)^{1/3} \times (3)^{2/3} \]
This can be rewritten as:
\[ \left(30 \times 100 \times 3^2\right)^{1/3} \]

### Step 8: Calculate \(30 \times 100 \times 3^2\)
Calculating \(3^2\):
\[ 3^2 = 9 \]
Now calculate:
\[ 30 \times 100 = 3000 \]
Then multiply:
\[ 3000 \times 9 = 27000 \]

### Step 9: Take the cube root
Now we find:
\[ (27000)^{1/3} = 30 \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{30} \]