To find the value of the expression [tex]\(\frac{1}{4}\left(c^3 + d^2\right)\)[/tex] when [tex]\(c = -4\)[/tex] and [tex]\(d = 10\)[/tex], follow these steps:
1. Substitute the given values into the expression:
[tex]\[
\frac{1}{4} \left((-4)^3 + 10^2\right)
\][/tex]
2. Calculate each part inside the parentheses:
- Find [tex]\((-4)^3\)[/tex] (the cube of -4):
[tex]\[
(-4)^3 = -64
\][/tex]
- Find [tex]\(10^2\)[/tex] (the square of 10):
[tex]\[
10^2 = 100
\][/tex]
3. Add the results from inside the parentheses:
[tex]\[
-64 + 100 = 36
\][/tex]
4. Multiply the sum by [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\frac{1}{4} \times 36 = 9
\][/tex]
So, the value of the expression [tex]\(\frac{1}{4}\left(c^3 + d^2\right)\)[/tex] when [tex]\(c = -4\)[/tex] and [tex]\(d = 10\)[/tex] is [tex]\(9\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{9} \][/tex]
