If `a + b = 10 and ab = 24,` then the value of `a ^(3) + b ^(3)` is

Options

A. 280
B. 370
C. 520
D. 730

Answer

To find the value of \( a^3 + b^3 \) given that \( a + b = 10 \) and \( ab = 24 \), we can use the identity:

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

### Step 1: Calculate \( a^2 + b^2 \)

We know that:

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

Substituting the known values:

\[ a + b = 10 \quad \text{and} \quad ab = 24 \]

We can calculate:

\[ a^2 + b^2 = (10)^2 - 2 \times 24 \]

Calculating \( (10)^2 \):

\[ = 100 \]

Calculating \( 2 \times 24 \):

\[ = 48 \]

Now substituting back:

\[ a^2 + b^2 = 100 - 48 = 52 \]

### Step 2: Substitute into the formula for \( a^3 + b^3 \)

Now we substitute \( a^2 + b^2 \) and \( ab \) into the formula for \( a^3 + b^3 \):

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

Substituting the known values:

\[ a^3 + b^3 = 10 \left( 52 - 24 \right) \]

Calculating \( 52 - 24 \):

\[ = 28 \]

Now substituting back:

\[ a^3 + b^3 = 10 \times 28 = 280 \]

### Final Answer

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

\[ \boxed{280} \]

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