Question

If the value of a-b=3 and $ {{a}^{3}}-{{b}^{3}}=117 $ , find the value of a+b?
(a) 5
(b) 7
(c) 9
(d) 11

Answer 1

Hint: We have values of a-b and \[{{a}^{3}}-{{b}^{3}}\] given in the problem. We use the formula $ {{a}^{3}}-{{b}^{3}}=\left( a-b \right)\times \left( {{a}^{2}}+{{b}^{2}}+ab \right) $ to proceed through the problem. After substituting the values of a-b and \[{{a}^{3}}-{{b}^{3}}\], We use $ {{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab $ to convert the equation in terms of $ {{\left( a+b \right)}^{2}} $ and ab. Now we apply cubing (power 3) to a-b on both sides to find the value of ab. After finding the value of ab, we substitute value in the previously obtained equation to find the value of a+b.

Complete step-by-step answer:
Given that the values of a-b=3 and $ {{a}^{3}}-{{b}^{3}}=117 $ . We need to find the value of a+b.
We know that $ {{a}^{3}}-{{b}^{3}}=\left( a-b \right)\times \left( {{a}^{2}}+{{b}^{2}}+ab \right) $ .
Now, we substitute the values of a-b=3 and $ {{a}^{3}}-{{b}^{3}}=117 $ that are already given in the problem.
 $ 117=3\times \left( {{a}^{2}}+{{b}^{2}}+ab \right) $ .
 $ {{a}^{2}}+{{b}^{2}}+ab=\dfrac{117}{3} $ .
 $ {{a}^{2}}+{{b}^{2}}+ab=39 $ .
Now, we add and subtract ab on L.H.S (Left Hand Side).
 $ {{a}^{2}}+{{b}^{2}}+ab+ab-ab=39 $
 $ {{a}^{2}}+{{b}^{2}}+2ab-ab=39 $ .
We know that $ {{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab $ .
 $ {{(a+b)}^{2}}-ab=39 $ ---(1).
Now, we have a-b=3.
Cubing on both sides, we get
 $ {{\left( a-b \right)}^{3}}={{3}^{3}} $ .
We know that $ {{\left( a-b \right)}^{3}}={{a}^{3}}-3{{a}^{2}}b+3a{{b}^{2}}-{{b}^{3}} $ .
 $ {{a}^{3}}-3{{a}^{2}}b+3a{{b}^{2}}-{{b}^{3}}=27 $ .
 $ {{a}^{3}}-{{b}^{3}}-3ab\left( a-b \right)=27 $ .
We have a-b=3 and $ {{a}^{3}}-{{b}^{3}}=117 $ . On substitute these values, we get
 $ 117-3ab\left( 3 \right)=27 $ .
 $ 117-27=9ab $ .
 $ 9ab=90 $ .
 $ ab=\dfrac{90}{9} $ .
ab = 10 ---(2).
Using equation (2) in equation (1), we get
 $ {{\left( a+b \right)}^{2}}-10=39 $ .
 $ {{\left( a+b \right)}^{2}}=39+10 $ .
 $ {{\left( a+b \right)}^{2}}=49 $ .
 $ a+b=\sqrt{49} $ .
a+b = 7.
∴ The value of a+b is 7.
So, the correct answer is “Option B”.

Note: We have taken the value of a+b is ‘7’, as the given options consist only of positive values. If there is an option with ‘-7’ in it, we could have also chosen that. If calculation mistakes are avoided, getting an answer to the solution is not a tough part. Similarly, we can also expect the problems to find the value of $ {{a}^{2}}+{{b}^{2}} $ .