To find the sum of the cubes of the first 19 natural numbers, we need to compute:
[tex]\[ 1^3 + 2^3 + 3^3 + 4^3 + \ldots + 19^3 \][/tex]
Step-by-step, we can methodically compute each term:
[tex]\[ 1^3 = 1 \][/tex]
[tex]\[ 2^3 = 8 \][/tex]
[tex]\[ 3^3 = 27 \][/tex]
[tex]\[ 4^3 = 64 \][/tex]
[tex]\[ 5^3 = 125 \][/tex]
[tex]\[ 6^3 = 216 \][/tex]
[tex]\[ 7^3 = 343 \][/tex]
[tex]\[ 8^3 = 512 \][/tex]
[tex]\[ 9^3 = 729 \][/tex]
[tex]\[ 10^3 = 1000 \][/tex]
[tex]\[ 11^3 = 1331 \][/tex]
[tex]\[ 12^3 = 1728 \][/tex]
[tex]\[ 13^3 = 2197 \][/tex]
[tex]\[ 14^3 = 2744 \][/tex]
[tex]\[ 15^3 = 3375 \][/tex]
[tex]\[ 16^3 = 4096 \][/tex]
[tex]\[ 17^3 = 4913 \][/tex]
[tex]\[ 18^3 = 5832 \][/tex]
[tex]\[ 19^3 = 6859 \][/tex]
Now, add up all these values to find the sum:
[tex]\[
1 + 8 + 27 + 64 + 125 + 216 + 343 + 512 + 729 + 1000 + 1331 + 1728 + 2197 + 2744 + 3375 + 4096 + 4913 + 5832 + 6859
\][/tex]
Calculating this sum:
[tex]\[
1 + 8 = 9 \\
9 + 27 = 36 \\
36 + 64 = 100 \\
100 + 125 = 225 \\
225 + 216 = 441 \\
441 + 343 = 784 \\
784 + 512 = 1296 \\
1296 + 729 = 2025 \\
2025 + 1000 = 3025 \\
3025 + 1331 = 4356 \\
4356 + 1728 = 6084 \\
6084 + 2197 = 8281 \\
8281 + 2744 = 11025 \\
11025 + 3375 = 14400 \\
14400 + 4096 = 18496 \\
18496 + 4913 = 23409 \\
23409 + 5832 = 29241 \\
29241 + 6859 = 36100
\][/tex]
Thus, the sum of the cubes of the first 19 natural numbers is:
[tex]\[ \boxed{36100} \][/tex]
