Question

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

Answer 1

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 \[{{(a-b)}^{2}}+2ab\].

Complete step by step answer:
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.