Question

If $a + b = 10$ and $ab = 21$, find the value of ${a^3} + {b^3}$.

Answer 1

Hint: In order to solve the problem, try to relate the terms with some algebraic identity. Look for some algebraic identity where the values given in the question can be directly substituted.

Complete step-by-step answer:

Given that $a + b = 10$ and $ab = 21$
We have to find ${a^3} + {b^3}$
As we know that
${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$
Now we will be putting the given values in the above equation directly
$
   \Rightarrow {\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right) \\
   \Rightarrow {\left( {10} \right)^3} = {a^3} + {b^3} + 3 \times 21\left( {10} \right) \\
   \Rightarrow 1000 = {a^3} + {b^3} + 630 \\
 $
Now let us bring the ${a^3} + {b^3}$ term on the left side and solve for it.
$
   \Rightarrow {a^3} + {b^3} = 1000 - 630 \\
   \Rightarrow {a^3} + {b^3} = 370 \\
 $
Hence the value of the given term ${a^3} + {b^3}$ is 370.

Note: The problem can also be solved by finding the individual values of a and b from the two given equations and then substituting the value found out in the problem term. But that method would have taken extra time and steps. So always try to relate this type of algebraic problem with some identities