Question

If $x - \dfrac{1}{x} = 3,$find the value of ${x^3} - \dfrac{1}{{{x^3}}}$

Answer 1

Hint: Take the given expression and take the whole cube of the terms as the required value is in the cube , use the identity of the whole cube and simplify placing the given values and simplify for the resultant required value.

Complete step-by-step answer:
Take the given expression: $x - \dfrac{1}{x} = 3$
Take whole cube on both the sides of the equation –
${\left( {x - \dfrac{1}{x}} \right)^3} = {3^3}$
Use the whole cube for the difference of the two terms and use the identity - ${(a - b)^3} = {a^3} - {b^3} - 3ab(a - b)$ and place the value for the cube of the term on the right hand side of the equation. When any term is multiplied thrice then we get the cube of its number.
${x^3} - {\left( {\dfrac{1}{x}} \right)^3} - 3(x)\left( {\dfrac{1}{x}} \right)\left( {x - \dfrac{1}{x}} \right) = {3^3}$
Common multiples from the numerator and the denominator cancels each other.
${x^3} - \left( {\dfrac{1}{{{x^3}}}} \right) - 3\left( {x - \dfrac{1}{x}} \right) = 27$
Place the given value for $x - \dfrac{1}{x} = 3,$in the above expression –
${x^3} - \left( {\dfrac{1}{{{x^3}}}} \right) - 3\left( 3 \right) = 27$
Simplify the above expression finding the product of the terms –
${x^3} - \left( {\dfrac{1}{{{x^3}}}} \right) - 9 = 27$
Make the required term the subject and move the other term on the opposite side. When you move any term from one side to the opposite side then the sign of the term is also changed. Negative term becomes positive and vice-versa.
${x^3} - \left( {\dfrac{1}{{{x^3}}}} \right) = 27 + 9$
Simplify the above expression, adding the terms on the right hand side of the equation –
${x^3} - \left( {\dfrac{1}{{{x^3}}}} \right) = 36$
The above expression is the required solution.
So, the correct answer is “36”.

Note: Remember the difference and sum of the terms and its whole cube and be careful about the sign convention. Always remember the sign convention while moving any term from one side to the other. When you move any term from one side to the opposite then the sign of the terms also changes. Algebraic identities of cube functions is very important to solve this question.