2025/020) Given $ab = 10$ and $a- b= 3$ find $a^3-b^3$

We can solve the 2 equations and get $a=5$ and $b=2$ and hence $a^3-b^3=5^3-2^3=117$

but the above approach is longer as compared to the approach below

We have

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

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

Putting the values we get $a^3-b^3=3^3 + 3 * 10 * 3 = 117$

In this process we have avoided solving of equations