Question

Find the value of $ {x^3} - \dfrac{1}{{{x^3}}} $ , if it is given that $ x - \dfrac{1}{x} = 5 $ .

Answer 1

Hint: In this problem, we have given that $ x - \dfrac{1}{x} = 5 $ and we have to find the value of $ {x^3} - \dfrac{1}{{{x^3}}} $ and to find this, firstly we need to square both sides of the given equation and then using the identity second that is given below, we will substitute the terms that we have found from the squaring of the equation.
Formula Used: We have used the two identities that are given below,
\[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\]
 $ {a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right) $

Complete step-by-step answer:
We have given an equation $ x - \dfrac{1}{x} = 5 $ and in order to find the value of $ {x^3} - \dfrac{1}{{{x^3}}} $ , we need to do squaring on both sides,
 $ {\left( {x - \dfrac{1}{x}} \right)^2} = {5^2} $
Now, we will solve this by substituting the values in first identity , $ x $ in place of $ a $ and $ \dfrac{1}{x} $ in place of b. After substituting, we get,
 $ {x^2} + \dfrac{1}{{{x^2}}} - 2 \times x \times \dfrac{1}{x} = 25 $
Now, on further solving, we get,
 $
   \Rightarrow {x^2} + \dfrac{1}{{{x^2}}} - 2 = 25 \\
   \Rightarrow \;{x^2} + \dfrac{1}{{{x^2}}} = 25 + 2 \\
   \Rightarrow {x^2} + \dfrac{1}{{{x^2}}} = 27 \;
  $
Now, we will elaborate the expression $ {x^3} - \dfrac{1}{{{x^3}}} $ by using the second identity by substituting $ x $ in place of $ a $ and $ \dfrac{1}{x} $ in place of b. Now, the expression becomes,
 $ {x^3} - \dfrac{1}{{{x^3}}} = \left( {x - \dfrac{1}{x}} \right)\left( {{x^2} + \dfrac{1}{{{x^2}}} + x \times \dfrac{1}{x}} \right) $
On further solving we get,
 $ {x^3} - \dfrac{1}{{{x^3}}} = \left( {x - \dfrac{1}{x}} \right)\left( {{x^2} + \dfrac{1}{{{x^2}}} + 1} \right) $
Now, we will substitute the values in the above equation, as in the question given that $ x - \dfrac{1}{x} = 5 $ and we have also find that $ {x^2} + \dfrac{1}{{{x^2}}} = 27 $ . Now, on substituting we get,
 $ {x^3} - \dfrac{1}{{{x^3}}} = 5 \times 28 $
On further solving, we get,
 $ {x^3} - \dfrac{1}{{{x^3}}} = 140 $
Hence, the value of $ {x^3} - \dfrac{1}{{{x^3}}} $ is $ 140 $ .
So, the correct answer is “140”.

Note: These types of problems are solved by using the identities, the student needs to learn and remember these as we don’t know where these identities become helpful while solving a problem. We can also find the value of $ {x^2} + \dfrac{1}{{{x^2}}} $ , if we know the value , $ x - \dfrac{1}{x} = 5 $ and $ {x^3} - \dfrac{1}{{{x^3}}} = 140 $ by using the second identity.