If `x^(3) + y^(3) = 175 and x + y = 7`. Find the value of `x^(4) + y^(4)`.

Options

A. 961
B. 916
C. 964
D. 912

Answer

To solve the problem, we need to find the value of \(x^4 + y^4\) given that \(x^3 + y^3 = 175\) and \(x + y = 7\).

### Step-by-step Solution:

1. **Use the identity for the sum of cubes**:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
We can rewrite this as:
\[ x^3 + y^3 = (x + y)((x + y)^2 - 3xy) \]

2. **Substitute the known values**:
We know \(x + y = 7\) and \(x^3 + y^3 = 175\). Substituting these into the equation gives:
\[ 175 = 7((7)^2 - 3xy) \]
Simplifying this:
\[ 175 = 7(49 - 3xy) \]
Dividing both sides by 7:
\[ 25 = 49 - 3xy \]

3. **Solve for \(xy\)**:
Rearranging the equation:
\[ 3xy = 49 - 25 \]
\[ 3xy = 24 \]
\[ xy = 8 \]

4. **Find \(x^2 + y^2\)**:
We can use the identity:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
Substituting the known values:
\[ x^2 + y^2 = 7^2 - 2 \cdot 8 \]
\[ x^2 + y^2 = 49 - 16 = 33 \]

5. **Use the identity for the sum of fourth powers**:
We can use the identity:
\[ x^4 + y^4 = (x^2 + y^2)^2 - 2(xy)^2 \]
Substituting the values we found:
\[ x^4 + y^4 = 33^2 - 2(8^2) \]
\[ x^4 + y^4 = 1089 - 2 \cdot 64 \]
\[ x^4 + y^4 = 1089 - 128 = 961 \]

### Final Answer:
Thus, the value of \(x^4 + y^4\) is \(961\).