The question is asking for the value of $ab$ under the given conditions:

1. $a + b = 5$
2. $a^3 + b^3 = 100$

Step-by-step Solution:

Step 1: Recall the identity for the sum of cubes:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Step 2: Substitute $a + b = 5$ into the equation:

$a^3 + b^3 = 5(a^2 - ab + b^2)$

Since $a^3 + b^3 = 100$, we can write: $100 = 5(a^2 - ab + b^2)$

Step 3: Simplify:

$a^2 - ab + b^2 = 20$

Step 4: Recall another identity for $a^2 + b^2$:

$a^2 + b^2 = (a + b)^2 - 2ab$

Substitute $a + b = 5$: $a^2 + b^2 = 5^2 - 2ab = 25 - 2ab$

Step 5: Substitute $a^2 + b^2 = 25 - 2ab$ into $a^2 - ab + b^2 = 20$:

$(25 - 2ab) - ab = 20$

Simplify: $25 - 3ab = 20$

$3ab = 5$

$ab = \frac{5}{3}$

Final Answer:

$ab = \frac{5}{3}$

Would you like a detailed explanation of any specific step?

Related Questions:

1. How do the sum and product of roots relate to the quadratic equation derived here?
2. Can you explain the derivation of the identity $a^2 + b^2 = (a+b)^2 - 2ab$?
3. What if $a + b \neq 5$? How would the problem change?
4. Could you solve this using a substitution method for $a$ and $b$?
5. How does this problem relate to symmetric polynomials?

Tip:

Using algebraic identities effectively can save significant time in problems involving powers of variables.