If `a +b= 11 and a^(2) + b^(2) = 65`, find `a^(3) + b^(3)`

Answer

To find the value of \( a^3 + b^3 \) given \( a + b = 11 \) and \( a^2 + b^2 = 65 \), we can use the identity:

\[ a^3 + b^3 = (a + b)(a^2 + b^2 - ab) \]

### Step 1: Find \( ab \)

We start with the equation \( a + b = 11 \). To find \( ab \), we will square both sides:

\[ (a + b)^2 = 11^2 \]

This expands to:

\[ a^2 + 2ab + b^2 = 121 \]

We know \( a^2 + b^2 = 65 \), so we can substitute this into the equation:

\[ 65 + 2ab = 121 \]

### Step 2: Solve for \( ab \)

Now, we can isolate \( 2ab \):

\[ 2ab = 121 - 65 \]
\[ 2ab = 56 \]

Dividing both sides by 2 gives us:

\[ ab = 28 \]

### Step 3: Substitute values into the identity

Now we have \( a + b = 11 \), \( a^2 + b^2 = 65 \), and \( ab = 28 \). We can substitute these values into the identity for \( a^3 + b^3 \):

\[ a^3 + b^3 = (a + b)(a^2 + b^2 - ab) \]

Substituting the known values:

\[ a^3 + b^3 = 11(65 - 28) \]

### Step 4: Calculate \( a^2 + b^2 - ab \)

Now we calculate \( 65 - 28 \):

\[ 65 - 28 = 37 \]

### Step 5: Final calculation

Now we can calculate \( a^3 + b^3 \):

\[ a^3 + b^3 = 11 \times 37 \]

Calculating this gives:

\[ a^3 + b^3 = 407 \]

### Final Answer

Thus, the value of \( a^3 + b^3 \) is \( \boxed{407} \).

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