QUESTION

# If $a-b=-8$ and $ab=-12$ then what is the value of ${{a}^{3}}-{{b}^{3}}$?A.-244B.-240C.-224D.-260

Hint:${{a}^{3}}-{{b}^{3}}$ can be written as $(a-b)({{a}^{2}}+ab+{{b}^{2}})$. Again ${{a}^{2}}+{{b}^{2}}$ can be written as ${{(a-b)}^{2}}+2ab$.

From the question it is given that $a-b=-8$ and $ab=-12$. Therefore, to find the value of ${{a}^{3}}-{{b}^{3}}$ we will first decompose it into factors. From basic algebra principles, we know the process and result of this very common algebraic expression.
We know that ${{a}^{3}}-{{b}^{3}}$ can be written as $(a-b)({{a}^{2}}+ab+{{b}^{2}})$. Now we know the values of $(a-b)$ and $ab$. However, the value of ${{a}^{2}}+{{b}^{2}}$ is still unknown. So, we have to find the value of ${{a}^{2}}+{{b}^{2}}$.
Again, this is another common algebraic expression and can be derived from basic algebra rules. Now, we know that ${{a}^{2}}+{{b}^{2}}={{a}^{2}}+{{b}^{2}}-2ab+2ab={{(a-b)}^{2}}+2ab$.
So, from the given values, putting these on the above equation we get,${{a}^{2}}+{{b}^{2}}={{(-8)}^{2}}+2(-12)=64-24=40$
So, now we know the value of ${{a}^{2}}+{{b}^{2}}$. Hence, we can easily find out our desired result. Now we will put altogether these values in the asked question.
Hence,${{a}^{3}}-{{b}^{3}}=(a-b)({{a}^{2}}+ab+{{b}^{2}})=(-8)(40-12)=28\times (-8)=-224$.
Hence, we arrived at our desired result. The value of ${{a}^{3}}-{{b}^{3}}$ is equal to then -224.
Therefore, the correct answer to the given question is option (c) -244.

Note: It is not mandatory or even necessary to get the values of a and b separately to finally get the answer. It is just needed to decompose the actual problem and get the desired result from the given data.