If `a+ b=5 and ab= 6`, find `a^3 + b^3`.

Answer

To find \( a^3 + b^3 \) given that \( a + b = 5 \) and \( ab = 6 \), we can use the identity:

\[ a^3 + b^3 = (a + b)^3 - 3ab(a + b) \]

### Step-by-Step Solution:

1. **Identify the values**:
We have:
\[ a + b = 5 \]
\[ ab = 6 \]

2. **Calculate \( (a + b)^3 \)**:
Using the value of \( a + b \):
\[ (a + b)^3 = 5^3 = 125 \]

3. **Calculate \( 3ab(a + b) \)**:
First, find \( 3ab \):
\[ 3ab = 3 \times 6 = 18 \]
Now multiply by \( a + b \):
\[ 3ab(a + b) = 18 \times 5 = 90 \]

4. **Substitute into the identity**:
Now substitute back into the identity:
\[ a^3 + b^3 = (a + b)^3 - 3ab(a + b) \]
\[ a^3 + b^3 = 125 - 90 \]

5. **Calculate the final result**:
\[ a^3 + b^3 = 35 \]

Thus, the value of \( a^3 + b^3 \) is \( 35 \).