Question

If $ a-b=7 $ and $ {{a}^{3}}-{{b}^{3}}=133 $ , then find the value of $ {{a}^{2}}+{{b}^{2}} $ ?

Answer 1

Hint: We start solving the problem by cubing the both sides of the given equation $ a-b=7 $ and make use of the result $ {{\left( a-b \right)}^{3}}={{a}^{3}}-{{b}^{3}}-3ab\left( a-b \right) $ . We then make the necessary calculations to get the value of $ ab $ . We then make use of the result $ {{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+{{b}^{2}}+ab \right) $ and we substitute the values of $ a-b $ , $ {{a}^{3}}-{{b}^{3}} $ and $ ab $ in the result to get the required answer.

Complete step by step answer:
According to the problem, we are given that $ a-b=7 $ and $ {{a}^{3}}-{{b}^{3}}=133 $ , and we need to find the value of $ {{a}^{2}}+{{b}^{2}} $ .
We have $ a-b=7 $ ---(1).
Let us apply cube on both sides of the equation (1).
So, we get $ {{\left( a-b \right)}^{3}}={{7}^{3}} $ ---(2).
We know that $ {{\left( a-b \right)}^{3}}={{a}^{3}}-{{b}^{3}}-3ab\left( a-b \right) $ . Let us use this in equation (2).
So, we get $ {{a}^{3}}-{{b}^{3}}-3ab\left( a-b \right)=343 $ ---(3).
Let us substitute the results $ a-b=7 $ and $ {{a}^{3}}-{{b}^{3}}=133 $ in equation (3).
So, we get $ 133-3ab\left( 7 \right)=343 $ .
 $ \Rightarrow -21ab=210 $ .
 $ \Rightarrow ab=-10 $ ---(4).
Now, we know that $ {{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+{{b}^{2}}+ab \right) $ .
But we already have $ a-b=7 $ , $ {{a}^{3}}-{{b}^{3}}=133 $ and $ ab=-10 $ .
 $ \Rightarrow 133=7\left( {{a}^{2}}+{{b}^{2}}-10 \right) $ .
 $ \Rightarrow 19=\left( {{a}^{2}}+{{b}^{2}}-10 \right) $ .
 $ \Rightarrow {{a}^{2}}+{{b}^{2}}=29 $ .
 $ \therefore $ We have found the value of $ {{a}^{2}}+{{b}^{2}} $ as 29.

Note:
We can see that the given problem contains a huge amount of calculation so, we need to perform each step carefully in order to avoid confusion and calculation mistakes. We can also solve the problem as shown below after finding the value of $ ab $ .
We know that $ {{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab $ . Let us substitute the results $ a-b=7 $ and $ ab=-10 $ .
 $ \Rightarrow {{\left( 7 \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2\left( -10 \right) $ .
 $ \Rightarrow 49={{a}^{2}}+{{b}^{2}}+20 $ .
 $ \Rightarrow {{a}^{2}}+{{b}^{2}}=29 $ . Similarly, we can expect problems to find the values of a, b, $ a+b $ and $ {{a}^{2}}-{{b}^{2}} $ .