Find the sum of `24(x^(2)-2x^(3))" and "-3(xy^(2)+y^(3))` and evaluate their sum at `x= (1)/(2)" and "y= 2`.

Answer

To solve the problem, we need to find the sum of the two algebraic expressions and then evaluate that sum at given values of \( x \) and \( y \).

### Step 1: Write down the expressions
The two expressions we need to sum are:
1. \( 24(x^2 - 2x^3) \)
2. \( -3(xy^2 + y^3) \)

### Step 2: Expand the expressions
We will first expand both expressions:
1. For \( 24(x^2 - 2x^3) \):
\[ 24x^2 - 48x^3 \]
2. For \( -3(xy^2 + y^3) \):
\[ -3xy^2 - 3y^3 \]

### Step 3: Combine the expanded expressions
Now, we combine the two expanded expressions:
\[ 24x^2 - 48x^3 - 3xy^2 - 3y^3 \]

### Step 4: Substitute the values of \( x \) and \( y \)
We need to evaluate the expression at \( x = \frac{1}{2} \) and \( y = 2 \):
\[ 24\left(\frac{1}{2}\right)^2 - 48\left(\frac{1}{2}\right)^3 - 3\left(\frac{1}{2}\right)(2^2) - 3(2^3) \]

### Step 5: Calculate each term
1. Calculate \( 24\left(\frac{1}{2}\right)^2 \):
\[ 24 \cdot \frac{1}{4} = 6 \]
2. Calculate \( -48\left(\frac{1}{2}\right)^3 \):
\[ -48 \cdot \frac{1}{8} = -6 \]
3. Calculate \( -3\left(\frac{1}{2}\right)(2^2) \):
\[ -3 \cdot \frac{1}{2} \cdot 4 = -6 \]
4. Calculate \( -3(2^3) \):
\[ -3 \cdot 8 = -24 \]

### Step 6: Combine all the results
Now we combine all the calculated terms:
\[ 6 - 6 - 6 - 24 \]

### Step 7: Simplify the expression
Combine the terms:
\[ 6 - 6 = 0 \\ 0 - 6 = -6 \\ -6 - 24 = -30 \]

Thus, the final result is:
\[ \text{The sum evaluated at } x = \frac{1}{2} \text{ and } y = 2 \text{ is } -30. \]