If `x^(2) + 4y^(2) + z^(2) - 4x - 2z + 5 = 0`, then find the value of `(y^(10) + x^(5) )/( z^(14) )`

Options

A. 32
B. 46
C. 58
D. 60

Answer

To solve the equation \( x^2 + 4y^2 + z^2 - 4x - 2z + 5 = 0 \) and find the value of \( \frac{y^{10} + x^5}{z^{14}} \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ x^2 + 4y^2 + z^2 - 4x - 2z + 5 = 0 \]
We will group the terms related to \( x \), \( y \), and \( z \).

### Step 2: Completing the Square for \( x \)
For the \( x \) terms:
\[ x^2 - 4x \]
To complete the square, we take half of the coefficient of \( x \) (which is -4), square it, and add/subtract it:
\[ x^2 - 4x = (x - 2)^2 - 4 \]

### Step 3: Completing the Square for \( y \)
For the \( y \) terms:
\[ 4y^2 = (2y)^2 \]
This is already a perfect square.

### Step 4: Completing the Square for \( z \)
For the \( z \) terms:
\[ z^2 - 2z \]
Completing the square:
\[ z^2 - 2z = (z - 1)^2 - 1 \]

### Step 5: Substitute Back into the Equation
Substituting back into the equation gives:
\[ (x - 2)^2 - 4 + (2y)^2 + (z - 1)^2 - 1 + 5 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (2y)^2 + (z - 1)^2 = 0 \]

### Step 6: Analyzing the Equation
Since a sum of squares equals zero, each square must be zero:
1. \( (x - 2)^2 = 0 \) → \( x - 2 = 0 \) → \( x = 2 \)
2. \( (2y)^2 = 0 \) → \( 2y = 0 \) → \( y = 0 \)
3. \( (z - 1)^2 = 0 \) → \( z - 1 = 0 \) → \( z = 1 \)

### Step 7: Substitute Values into the Expression
Now we substitute \( x = 2 \), \( y = 0 \), and \( z = 1 \) into the expression:
\[ \frac{y^{10} + x^5}{z^{14}} = \frac{0^{10} + 2^5}{1^{14}} \]
Calculating this gives:
\[ \frac{0 + 32}{1} = 32 \]

### Final Answer
Thus, the value of \( \frac{y^{10} + x^5}{z^{14}} \) is:
\[ \boxed{32} \]