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If $x - \dfrac{1}{x} = - \sqrt 3$, then find ${x^3} - \dfrac{1}{{{x^3}}}$A) $6\sqrt 3$B) $2\sqrt 3$C) $3\sqrt 3$D) $- 6\sqrt 3$

Last updated date: 22nd Jun 2024
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Hint:
Here we will take the cube on both sides of the given equation and then we will apply the formula of the cube of difference. We will simplify the equation further and from there, we will get the value of the required expression.

Formula Used:
We will use the formula of cube of difference which is given by ${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)$.

Complete step by step solution:
It is given that $x - \dfrac{1}{x} = - \sqrt 3$.
Now, taking cube on both sides, we get
$\Rightarrow {\left( {x - \dfrac{1}{x}} \right)^3} = {\left( { - \sqrt 3 } \right)^3}$
Applying the exponent on the term, we get
$\Rightarrow {\left( {x - \dfrac{1}{x}} \right)^3} = - 3\sqrt 3$
Now, using the formula ${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)$ in the above equation, we get
$\Rightarrow {x^3} - \dfrac{1}{{{x^3}}} - 3x \cdot \dfrac{1}{x} \cdot \left( {x - \dfrac{1}{x}} \right) = - 3\sqrt 3$
Simplifying the above equation, we get
$\Rightarrow {x^3} - \dfrac{1}{{{x^3}}} - 3 \cdot \left( {x - \dfrac{1}{x}} \right) = - 3\sqrt 3$
We know the value $x - \dfrac{1}{x} = - \sqrt 3$. Now, we will substitute this value in the above equation. Therefore, we get
$\Rightarrow {x^3} - \dfrac{1}{{{x^3}}} - 3 \cdot \left( { - \sqrt 3 } \right) = - 3\sqrt 3$
On multiplying the terms, we get
$\Rightarrow {x^3} - \dfrac{1}{{{x^3}}} + 3\sqrt 3 = - 3\sqrt 3$
On subtracting $3\sqrt 3$ on both sides, we get
$\Rightarrow {x^3} - \dfrac{1}{{{x^3}}} + 3\sqrt 3 - 3\sqrt 3 = - 3\sqrt 3 - 3\sqrt 3$
$\Rightarrow {x^3} - \dfrac{1}{{{x^3}}} = - 6\sqrt 3$

Hence, the correct option is option D.

Note:
Here, we have used algebraic identity to solve the given algebraic equation. We generally use the algebraic identities for factorization of polynomials. But we should remember that algebraic expressions and algebraic identities are different. An algebraic expression is defined as an expression which consists of constants and variables. In algebraic expressions, a variable can take any value. Thus, the expression value can change if the variable values are changed. But algebraic identity is defined as an equality which is true for all the values of the variables.
Here, we might make a mistake by writing ${\left( { - \sqrt 3 } \right)^3}$ as $3\sqrt 3$ and forgot the minus sign. When a negative number is multiplied three times then the resulting number will be a negative and not positive.