Question

If $a + b = 8$ and $ab = 6$ , find the value of ${a^3} + {b^3}$

Answer 1

Hint: We can substitute the given equations in the binomial expansion of ${\left( {a + b} \right)^3}$ . Then we can rearrange and simplify the equation to get the required answer.

Complete step by step solution:
Given $a + b = 8$ and $ab = 6$
When we look at the expression ${a^3} + {b^3}$ , we can see that they are the 1st and last terms of binomial expansion of ${\left( {a + b} \right)^3}$ . The expansion of ${\left( {a + b} \right)^3}$ is given by,
${\left( {a + b} \right)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3}$
Taking common terms $3ab$ from the middle terms, we get,
${\left( {a + b} \right)^3} = {a^3} + 3ab\left( {a + b} \right) + {b^3}$
On subtracting both sides with $3ab\left( {a + b} \right)$ , we get,
${a^3} + {b^3} = {\left( {a + b} \right)^3} - 3ab\left( {a + b} \right)$
From the question, we have, $a + b = 8$ and $ab = 6$
On substituting the values in above equation, we get,
$ \Rightarrow {a^3} + {b^3} = {\left( 8 \right)^3} - 3 \times 6 \times \left( 8 \right)$
On simplifying we get,
$ \Rightarrow {a^3} + {b^3} = 512 - 144$
$ \Rightarrow {a^3} + {b^3} = 368$
Therefore, the value of ${a^3} + {b^3}$ is 368.

Note: We use the concept of binomial expansion to solve the problem. Binomial expansion of any positive integer can be found using the equation
${\left( {a + b} \right)^n} = {a^n} + \left( {{}^n{c_1}} \right){a^{n - 1}}b + \left( {{}^n{c_2}} \right){a^{n - 2}}{b^2} + \ldots + \left( {{}^n{c_{n - 1}}} \right)a{b^{n - 1}} + {b^n}$
When $n = 3$ , we get,
${\left( {a + b} \right)^3} = {a^3} + \left( {{}^3{C_1}} \right){a^2}b + \left( {{}^3{C_2}} \right){a^1}{b^2} + {b^3}$
We know that $\left( {{}^3{C_1}} \right) = \left( {{}^3{C_2}} \right) = 3$
$ \Rightarrow {\left( {a + b} \right)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3}$
This the equation we used in the problem.