The sum of two numbers is 4 and their product is 3. Find : (ii) the sum of their cubes.

Answer

To find the sum of the cubes of two numbers given that their sum is 4 and their product is 3, we can follow these steps:

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \).
Given:
- \( x + y = 4 \) (sum of the numbers)
- \( xy = 3 \) (product of the numbers)

### Step 2: Use the Identity for the Sum of Cubes
We know the identity for the sum of cubes:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
We can also express \( x^2 + y^2 \) in terms of \( x + y \) and \( xy \):
\[ x^2 + y^2 = (x + y)^2 - 2xy \]

### Step 3: Calculate \( x^2 + y^2 \)
Using the values we have:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
Substituting the known values:
\[ x^2 + y^2 = (4)^2 - 2(3) \]
\[ x^2 + y^2 = 16 - 6 = 10 \]

### Step 4: Substitute into the Sum of Cubes Formula
Now we can substitute \( x + y \) and \( x^2 + y^2 \) into the sum of cubes formula:
\[ x^3 + y^3 = (x + y)((x^2 + y^2) - xy) \]
Substituting the values:
\[ x^3 + y^3 = 4 \left( 10 - 3 \right) \]
\[ x^3 + y^3 = 4 \times 7 = 28 \]

### Final Answer
The sum of the cubes of the two numbers is \( 28 \).

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