If (x+y)=8 and xy=15 then the value of `(x^3-y^3)` is :

Options

A. 120
B. 60
C. 98
D. 82

Answer

To find the value of \( x^3 - y^3 \) given that \( x + y = 8 \) and \( xy = 15 \), we can follow these steps:

### Step 1: Use the identity for \( x^3 - y^3 \)
We know that:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]
To use this identity, we need to find \( x - y \) and \( x^2 + xy + y^2 \).

### Step 2: Find \( x - y \)
We can find \( x - y \) using the identity:
\[ (x + y)^2 = x^2 + y^2 + 2xy \]
From the problem, we know \( x + y = 8 \) and \( xy = 15 \). Therefore:
\[ (x + y)^2 = 8^2 = 64 \]
Substituting into the equation:
\[ 64 = x^2 + y^2 + 2 \cdot 15 \]
This simplifies to:
\[ 64 = x^2 + y^2 + 30 \]
Thus:
\[ x^2 + y^2 = 64 - 30 = 34 \]

Now we can find \( x - y \) using the identity:
\[ (x - y)^2 = x^2 + y^2 - 2xy \]
Substituting the known values:
\[ (x - y)^2 = 34 - 2 \cdot 15 = 34 - 30 = 4 \]
Taking the square root:
\[ x - y = \sqrt{4} = 2 \]

### Step 3: Find \( x^2 + xy + y^2 \)
Now we can calculate \( x^2 + xy + y^2 \):
\[ x^2 + xy + y^2 = x^2 + y^2 + xy = 34 + 15 = 49 \]

### Step 4: Substitute into the identity
Now we can substitute \( x - y \) and \( x^2 + xy + y^2 \) into the identity:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) = 2 \cdot 49 = 98 \]

### Final Answer
Thus, the value of \( x^3 - y^3 \) is:
\[ \boxed{98} \]