Answer

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**Hint**: For solving this type of question we always have to frame the equation in such a way that it will follow the formula which is given by $ {\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab $ . In this we will add $ 2 $ both the sides and framing the equation, we get the value for $ {x^3} + \dfrac{1}{{{x^3}}} $ .

Formula used:

The algebraic formula will be given by

$ {\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab $

Here, $ a\& b $ will be the variables.

**:**

__Complete step-by-step answer__We have the equation given as $ {x^4} + \dfrac{1}{{{x^4}}} = 194 $

Now on adding both the sides $ 2 $ , we get the equation as

$ \Rightarrow {x^4} + \dfrac{1}{{{x^4}}} + 2 = 194 + 2 $

Now by using the formula, the left side of the equation will be given as

$ \Rightarrow {\left( {{x^2} + \dfrac{1}{{{x^2}}}} \right)^2} = 196 $

By removing the square from the left sides of the equation, we will get the equation as

$ \Rightarrow \left( {{x^2} + \dfrac{1}{{{x^2}}}} \right) = 14 $

Now again adding $ 2 $ both the sides of the equation, we will get

$ \Rightarrow {x^2} + \dfrac{1}{{{x^2}}} + 2 = 14 + 2 $

Again we can see that the left side of the equation is following the formula $ {\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab $

$ \Rightarrow {\left( {x + \dfrac{1}{x}} \right)^2} = 16 $

By removing the square from the left sides of the equation, we will get the equation as

$ \Rightarrow x + \dfrac{1}{x} = 4 $ , and we will name it equation $ 1 $

Now for finding the $ {x^3} + \dfrac{1}{{{x^3}}} $ , we will take out the cube root of the equation $ 1 $ .

$ \Rightarrow {\left( {x + \dfrac{1}{x}} \right)^3} = {4^3} $

We get,

\[ \Rightarrow {\left( {x + \dfrac{1}{x}} \right)^3} = {x^3} + \dfrac{1}{{{x^3}}} + 3\left( {x + \dfrac{1}{x}} \right)\]

And from this, we will get

$ \Rightarrow {4^3} = {x^3} + \dfrac{1}{{{x^3}}} +3 \times 4 $

And on solving and taking the constant term to one side, we get the equation as

$ \Rightarrow {x^3} + \dfrac{1}{{{x^3}}} = 52 $

Hence, the option $ \left( b \right) $ is correct.

**So, the correct answer is “Option b”.**

**Note**: Here, in this question, we can see that with the help of the formula we can easily solve this type of question. So the important thing for us will be to memorize the formula and by practice, we will get the knowledge of substituting the formula with the question. Also while solving the question for the long answer question, we should follow each step.

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