If `x^(3) + y^(3)=35 and x+y=5` then find the value of `x^(4) + y^(4)`

Options

A. A)87
B. B)89
C. C)97
D. D)93

Answer

To solve the problem, we need to find the value of \( x^4 + y^4 \) given that \( x^3 + y^3 = 35 \) and \( x + y = 5 \).

### Step-by-Step Solution:

1. **Use the identity for the sum of cubes**:
The identity for the sum of cubes is given by:
\[ x^3 + y^3 = (x+y)(x^2 - xy + y^2) \]
We know \( x + y = 5 \), so we can substitute this into the identity:
\[ 35 = 5(x^2 - xy + y^2) \]

2. **Simplify the equation**:
Dividing both sides by 5 gives:
\[ x^2 - xy + y^2 = 7 \]

3. **Use the square of the sum**:
We also know that:
\[ (x+y)^2 = x^2 + 2xy + y^2 \]
Substituting \( x + y = 5 \):
\[ 5^2 = x^2 + 2xy + y^2 \implies 25 = x^2 + 2xy + y^2 \]

4. **Express \( x^2 + y^2 \)**:
We can express \( x^2 + y^2 \) in terms of \( xy \):
\[ x^2 + y^2 = (x^2 - xy + y^2) + xy = 7 + xy \]
Thus:
\[ 25 = (7 + xy) + 2xy \implies 25 = 7 + 3xy \]

5. **Solve for \( xy \)**:
Rearranging gives:
\[ 3xy = 25 - 7 \implies 3xy = 18 \implies xy = 6 \]

6. **Find \( x^2 + y^2 \)**:
Now we can find \( x^2 + y^2 \):
\[ x^2 + y^2 = 7 + xy = 7 + 6 = 13 \]

7. **Use the identity for the sum of fourth powers**:
The identity for the sum of fourth powers is:
\[ x^4 + y^4 = (x^2 + y^2)^2 - 2(xy)^2 \]
Substituting the values we found:
\[ x^4 + y^4 = (13)^2 - 2(6)^2 \]

8. **Calculate**:
\[ x^4 + y^4 = 169 - 2 \times 36 = 169 - 72 = 97 \]

### Final Answer:
Thus, the value of \( x^4 + y^4 \) is \( \boxed{97} \).