To solve the expression [tex]\(\frac{1}{4}\left(c^3 + d^2\right)\)[/tex] when [tex]\(c = -4\)[/tex] and [tex]\(d = 10\)[/tex], we need to follow these steps:
1. Substitute the values of [tex]\(c\)[/tex] and [tex]\(d\)[/tex] into the expression.
- [tex]\(c = -4\)[/tex]
- [tex]\(d = 10\)[/tex]
2. Calculate [tex]\(c^3\)[/tex].
- We need to find [tex]\((-4)^3\)[/tex].
- [tex]\((-4)^3 = -4 \times -4 \times -4\)[/tex].
- [tex]\(-4 \times -4 = 16\)[/tex], and [tex]\(16 \times -4 = -64\)[/tex].
- Thus, [tex]\(c^3 = -64\)[/tex].
3. Calculate [tex]\(d^2\)[/tex].
- We need to find [tex]\(10^2\)[/tex].
- [tex]\(10^2 = 10 \times 10 = 100\)[/tex].
- Thus, [tex]\(d^2 = 100\)[/tex].
4. Add the results of [tex]\(c^3\)[/tex] and [tex]\(d^2\)[/tex].
- We now sum [tex]\(-64\)[/tex] and [tex]\(100\)[/tex].
- [tex]\(-64 + 100 = 36\)[/tex].
5. Multiply the sum by [tex]\(\frac{1}{4}\)[/tex].
- We need to find [tex]\(\frac{1}{4} \times 36\)[/tex].
- [tex]\(\frac{1}{4} \times 36 = 36 \div 4 = 9\)[/tex].
6. Determine the final result of the expression.
- The value of [tex]\(\frac{1}{4}\left(c^3 + d^2\right)\)[/tex] is [tex]\(9\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{9}
\][/tex]
