What is the value of `x^(4) + y^(4)` when the value of `x^(3) + y^(3) =8` and `x+y =2` ?

Options

A. 2
B. 8
C. 16
D. 32

Answer

To solve the problem, we need to find the value of \( x^4 + y^4 \) given that \( x^3 + y^3 = 8 \) and \( x + y = 2 \).

### Step 1: Use the identity for \( x^3 + y^3 \)
We know that:
\[ x^3 + y^3 = (x+y)(x^2 - xy + y^2) \]
Given \( x + y = 2 \), we can substitute this into the identity:
\[ x^3 + y^3 = 2(x^2 - xy + y^2) \]

### Step 2: Substitute the value of \( x^3 + y^3 \)
We are given \( x^3 + y^3 = 8 \), so we can set up the equation:
\[ 8 = 2(x^2 - xy + y^2) \]
Dividing both sides by 2 gives:
\[ 4 = x^2 - xy + y^2 \]

### Step 3: Use the identity for \( x^2 + y^2 \)
We can express \( x^2 + y^2 \) in terms of \( (x+y)^2 \) and \( xy \):
\[ x^2 + y^2 = (x+y)^2 - 2xy \]
Substituting \( x + y = 2 \):
\[ x^2 + y^2 = 2^2 - 2xy = 4 - 2xy \]

### Step 4: Substitute into the equation
Now we substitute \( x^2 + y^2 \) back into the equation we derived from \( x^3 + y^3 \):
\[ 4 = (4 - 2xy) - xy \]
This simplifies to:
\[ 4 = 4 - 3xy \]
Subtracting 4 from both sides gives:
\[ 0 = -3xy \]
Thus, we find:
\[ xy = 0 \]

### Step 5: Find \( x^2 + y^2 \)
Now we can find \( x^2 + y^2 \):
\[ x^2 + y^2 = 4 - 2(0) = 4 \]

### Step 6: Use the identity for \( x^4 + y^4 \)
We can use the identity:
\[ x^4 + y^4 = (x^2 + y^2)^2 - 2(xy)^2 \]
Substituting the values we found:
\[ x^4 + y^4 = (4)^2 - 2(0)^2 = 16 - 0 = 16 \]

### Final Answer
Thus, the value of \( x^4 + y^4 \) is:
\[ \boxed{16} \]