To solve the summation [tex]\(\sum_{k=4}^7 k^3\)[/tex], we need to evaluate each term in the summation and then sum them up.
The expression asks us to find the sum of the cubes of integers from 4 to 7. We will proceed step by step:
1. Evaluate each term in the summation:
- For [tex]\( k = 4 \)[/tex]:
[tex]\[
4^3 = 4 \times 4 \times 4 = 64
\][/tex]
- For [tex]\( k = 5 \)[/tex]:
[tex]\[
5^3 = 5 \times 5 \times 5 = 125
\][/tex]
- For [tex]\( k = 6 \)[/tex]:
[tex]\[
6^3 = 6 \times 6 \times 6 = 216
\][/tex]
- For [tex]\( k = 7 \)[/tex]:
[tex]\[
7^3 = 7 \times 7 \times 7 = 343
\][/tex]
2. Sum up these evaluated terms:
[tex]\[
64 + 125 + 216 + 343
\][/tex]
3. Compute the sum step-by-step:
- First, add [tex]\( 64 \)[/tex] and [tex]\( 125 \)[/tex]:
[tex]\[
64 + 125 = 189
\][/tex]
- Next, add the result to [tex]\( 216 \)[/tex]:
[tex]\[
189 + 216 = 405
\][/tex]
- Finally, add the last term [tex]\( 343 \)[/tex] to this result:
[tex]\[
405 + 343 = 748
\][/tex]
So, after summing up all the values, we get [tex]\( 748 \)[/tex].
Therefore, the value of [tex]\(\sum_{k=4}^7 k^3\)[/tex] is
[tex]\(\boxed{748}\)[/tex].
The correct answer is:
(D) 748
