If `(x+y)=13 and xy=36`, then the value of `(x^(3)+y^(3))` is :

Options

A. 369
B. 936
C. 693
D. none of these

Answer

To find the value of \( x^3 + y^3 \) given that \( x + y = 13 \) and \( xy = 36 \), we can use the identity for the sum of cubes:

\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]

### Step 1: Calculate \( x^2 + y^2 \)

We can find \( x^2 + y^2 \) using the identity:

\[ x^2 + y^2 = (x + y)^2 - 2xy \]

Substituting the known values:

\[ x^2 + y^2 = (13)^2 - 2(36) \]

Calculating this:

\[ x^2 + y^2 = 169 - 72 = 97 \]

### Step 2: Substitute into the sum of cubes formula

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

\[ x^3 + y^3 = (x + y)((x^2 + y^2) - xy) \]

Substituting the known values:

\[ x^3 + y^3 = 13(97 - 36) \]

Calculating the expression inside the parentheses:

\[ 97 - 36 = 61 \]

### Step 3: Final calculation

Now we can calculate \( x^3 + y^3 \):

\[ x^3 + y^3 = 13 \times 61 \]

Calculating this gives:

\[ x^3 + y^3 = 793 \]

Thus, the value of \( x^3 + y^3 \) is \( \boxed{793} \).

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