If `a+b =5 and ab = 3`, then `(a^(3)+b^(3))` is equal to :

Options

A. 70
B. 65
C. 75
D. 80

Answer

To solve the problem where \( a + b = 5 \) and \( ab = 3 \), we need to find \( a^3 + b^3 \). We can use the formula for the sum of cubes:

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

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

We can find \( a^2 + b^2 \) using the identity:

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

Substituting the known values:

\[ a^2 + b^2 = (5)^2 - 2(3) \]

Calculating this gives:

\[ a^2 + b^2 = 25 - 6 = 19 \]

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

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

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

Substituting the values we have:

\[ a^3 + b^3 = (5)(19 - 3) \]

Calculating this gives:

\[ a^3 + b^3 = 5(16) = 80 \]

### Final Answer

Thus, \( a^3 + b^3 = 80 \).

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