Simplify the equation `((a)/(2)-(b)/(3))^(3)+((a)/(2)+(b)/(3))^(3)`.

Options

A. `(a^(3))/(4)+ab^(2)`
B. `(a^(3))/(4)+(ab^(2))/(3)`
C. `(2b^(3))/(27)+(a^(2)b)/(2)`
D. `(2b^(3))/(27)+a^(2)b`

Answer

To simplify the expression \(\left(\frac{a}{2} - \frac{b}{3}\right)^3 + \left(\frac{a}{2} + \frac{b}{3}\right)^3\), we can use the identity for the sum of cubes. The identity states that:

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

where \(x = \frac{a}{2} - \frac{b}{3}\) and \(y = \frac{a}{2} + \frac{b}{3}\).

### Step 1: Identify \(x\) and \(y\)
Let:
\[ x = \frac{a}{2} - \frac{b}{3} \]
\[ y = \frac{a}{2} + \frac{b}{3} \]

### Step 2: Calculate \(x + y\)
Now, we find \(x + y\):
\[ x + y = \left(\frac{a}{2} - \frac{b}{3}\right) + \left(\frac{a}{2} + \frac{b}{3}\right) = \frac{a}{2} + \frac{a}{2} = a \]

### Step 3: Calculate \(xy\)
Next, we calculate \(xy\):
\[ xy = \left(\frac{a}{2} - \frac{b}{3}\right)\left(\frac{a}{2} + \frac{b}{3}\right) = \left(\frac{a}{2}\right)^2 - \left(\frac{b}{3}\right)^2 = \frac{a^2}{4} - \frac{b^2}{9} \]

### Step 4: Calculate \(x^2 + y^2\)
Now, we calculate \(x^2 + y^2\):
\[ x^2 + y^2 = \left(\frac{a}{2} - \frac{b}{3}\right)^2 + \left(\frac{a}{2} + \frac{b}{3}\right)^2 \]
Using the formula \( (u - v)^2 + (u + v)^2 = 2u^2 + 2v^2 \):
\[ = 2\left(\frac{a}{2}\right)^2 + 2\left(\frac{b}{3}\right)^2 = 2\left(\frac{a^2}{4}\right) + 2\left(\frac{b^2}{9}\right) = \frac{a^2}{2} + \frac{2b^2}{9} \]

### Step 5: Substitute into the sum of cubes formula
Now substituting into the sum of cubes formula:
\[ x^3 + y^3 = (x + y)\left(x^2 - xy + y^2\right) \]
\[ = a\left(\frac{a^2}{2} + \frac{2b^2}{9} - \left(\frac{a^2}{4} - \frac{b^2}{9}\right)\right) \]

### Step 6: Simplify the expression
Now simplify:
\[ = a\left(\frac{a^2}{2} + \frac{2b^2}{9} - \frac{a^2}{4} + \frac{b^2}{9}\right) \]
Combining the terms:
\[ = a\left(\frac{2a^2}{4} - \frac{a^2}{4} + \frac{3b^2}{9}\right) = a\left(\frac{a^2}{4} + \frac{b^2}{3}\right) \]

### Final Result
Thus, the simplified expression is:
\[ \frac{a}{4}\left(a^2 + \frac{4b^2}{3}\right) \]