Question

If $a-b=4,ab=45.$ Then find the value of ${{a}^{3}}-{{b}^{3}}$

Answer 1

Hint: Use the identity ${{\left( a-b \right)}^{3}}={{a}^{3}}-{{b}^{3}}-3ab(a-b)$ . We have the value of a-b and ab. We just need to put the value of that and we will get the required answer.

Complete step-by-step answer:
As we know
${{\left( a-b \right)}^{3}}={{a}^{3}}-{{b}^{3}}-3ab(a-b)$
We can write by transposing
${{a}^{3}}-{{b}^{3}}={{(a-b)}^{3}}+3ab(a-b)$
Now from question we have
$\begin{align}
  & a-b=4 \\
 & ab=45 \\
\end{align}$
So putting all the value in the above
$\begin{align}
  & {{a}^{3}}-{{b}^{3}}={{(4)}^{3}}+3(45)(4) \\
 & \Rightarrow {{a}^{3}}-{{b}^{3}}=64+12x45 \\
 & \Rightarrow {{a}^{3}}-{{b}^{3}}=64+540 \\
 & \Rightarrow {{a}^{3}}-{{b}^{3}}=604 \\
\end{align}$

Note: This question can also be solved by solving simultaneous equation, for this first we have to find the positive value of a+b by using the identity ${{\left( a+b \right)}^{2}}={{\left( a-b \right)}^{2}}+4ab$. By solving we get value of a and b, then find ${{a}^{3}}-{{b}^{3}}$