Find the value of ` sqrt (( x^(2) + y^(2) +z) ( x + y - 3 z))+ 3 sqrt( xy^(3) z^(2))` when ` x = - 1 , y = - 3, z = - 1 `

Options

A. (A) 1
B. (B) 0
C. (C) 2
D. (D) NONE OF THESE

Answer

To find the value of the expression \( \sqrt{(x^2 + y^2 + z)(x + y - 3z)} + 3\sqrt{xy^3z^2} \) when \( x = -1 \), \( y = -3 \), and \( z = -1 \), we will follow these steps:

### Step 1: Substitute the values of \( x \), \( y \), and \( z \)

Substituting \( x = -1 \), \( y = -3 \), and \( z = -1 \) into the expression:

\[ \sqrt{((-1)^2 + (-3)^2 + (-1)((-1) + (-3) - 3(-1)))} + 3\sqrt{(-1)(-3)^3(-1)^2} \]

### Step 2: Calculate \( x^2 + y^2 + z \)

Calculate \( x^2 + y^2 + z \):

\[ (-1)^2 = 1, \quad (-3)^2 = 9, \quad z = -1 \]

So,

\[ x^2 + y^2 + z = 1 + 9 - 1 = 9 \]

### Step 3: Calculate \( x + y - 3z \)

Calculate \( x + y - 3z \):

\[ -1 + (-3) - 3(-1) = -1 - 3 + 3 = -1 \]

### Step 4: Substitute into the square root expression

Now substitute these results back into the expression:

\[ \sqrt{(9)(-1)} + 3\sqrt{(-1)(-3)^3(-1)^2} \]

### Step 5: Calculate the first square root

Since \( \sqrt{(9)(-1)} \) involves the square root of a negative number, we have:

\[ \sqrt{-9} = 3i \quad (\text{where } i \text{ is the imaginary unit}) \]

### Step 6: Calculate \( 3\sqrt{(-1)(-3)^3(-1)^2} \)

Now calculate \( 3\sqrt{(-1)(-3)^3(-1)^2} \):

\[ (-3)^3 = -27, \quad (-1)^2 = 1 \]

So,

\[ (-1)(-27)(1) = 27 \]

Thus,

\[ 3\sqrt{27} = 3 \times 3\sqrt{3} = 9\sqrt{3} \]

### Step 7: Combine the results

Now combine the results:

\[ 3i + 9\sqrt{3} \]

### Final Answer

The final value of the expression is:

\[ 9\sqrt{3} + 3i \]