To find [tex]\( a^3 + b^3 \)[/tex] given the equations [tex]\( a + b = 10 \)[/tex] and [tex]\( a^2 + b^2 = 58 \)[/tex], follow these steps:
1. Identify the given information:
- [tex]\( a + b = 10 \)[/tex]
- [tex]\( a^2 + b^2 = 58 \)[/tex]
2. Use the identity for the square of a sum:
[tex]\[ (a + b)^2 = a^2 + b^2 + 2ab \][/tex]
3. Substitute the known values into the identity:
[tex]\[ 10^2 = 58 + 2ab \][/tex]
[tex]\[ 100 = 58 + 2ab \][/tex]
4. Solve for [tex]\( ab \)[/tex]:
[tex]\[ 100 = 58 + 2ab \][/tex]
[tex]\[ 100 - 58 = 2ab \][/tex]
[tex]\[ 42 = 2ab \][/tex]
[tex]\[ ab = 21 \][/tex]
5. Use the identity for the sum of cubes:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
6. Substitute the known values into this identity:
[tex]\[ a^3 + b^3 = (a + b)(a^2 + b^2 - ab) \][/tex]
[tex]\[ a^3 + b^3 = 10(58 - 21) \][/tex]
7. Simplify the expression:
[tex]\[ a^3 + b^3 = 10 \times 37 \][/tex]
[tex]\[ a^3 + b^3 = 370 \][/tex]
Thus, [tex]\( a^3 + b^3 = 370 \)[/tex].
