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Hint – In this question the value of x is given and we need to find the value of the given expression, take 2 on the left hand side towards x and take cube both the sides. Use the algebraic identity of ${\left( {a - b} \right)^3}$and others to reach the answer.

Complete step-by-step answer:

Given equation is

$x = 2 + {2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}}$

So, we have to find out the value of ${x^3} - 6{x^2} + 6x$.

Now in given equation take 2 to L.H.S and take cube on both sides we have,

$ \Rightarrow x - 2 = {2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}}$ ………………….. (1)

\[ \Rightarrow {\left( {x - 2} \right)^3} = {\left( {{2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}}} \right)^3}\]

Now as we know ${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}$ and ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}$ so, apply this property in above equation we have,

\[ \Rightarrow {x^3} - {2^3} - 3\left( {{x^2}} \right)\left( 2 \right) + 3x\left( {{2^2}} \right) = {\left( {{2^{\dfrac{2}{3}}}} \right)^3} + {\left( {{2^{\dfrac{1}{3}}}} \right)^3} + 3{\left( {{2^{\dfrac{2}{3}}}} \right)^2}\left( {{2^{\dfrac{1}{3}}}} \right) + 3\left( {{2^{\dfrac{2}{3}}}} \right){\left( {{2^{\dfrac{1}{3}}}} \right)^2}\]

Now simplify the above equation we have,

\[ \Rightarrow {x^3} - 8 - 6{x^2} + 12x = {2^2} + 2 + 3\left( {{2^{\dfrac{2}{3}}}} \right)\left( {{2^{\dfrac{1}{3}}}} \right)\left( {{2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}}} \right)\]

Now from equation (1) we have,

\[ \Rightarrow {x^3} - 8 - 6{x^2} + 12x = {2^2} + 2 + 3\left( {{2^{\dfrac{2}{3} + \dfrac{1}{3}}}} \right)\left( {x - 2} \right)\]

Now simplify the above equation we have,

\[ \Rightarrow {x^3} - 8 - 6{x^2} + 12x = 6 + \left( {3 \times 2\left( {x - 2} \right)} \right)\]

\[ \Rightarrow {x^3} - 8 - 6{x^2} + 12x = 6 + 6x - 12 = 6x - 6\]

\[ \Rightarrow {x^3} - 8 - 6{x^2} + 6x = - 6\]

\[ \Rightarrow {x^3} - 6{x^2} + 6x = 8 - 6 = 2\]

So the required value of ${x^3} - 6{x^2} + 6x$ is 2.

So, this is the required answer.

Note – Whenever we face such types of problems the key concept is simply not to substitute the value of x in the given expression but somehow to simply and to change the expression into a bigger expression containing sub expressions whose values are known to us. This concept will help you get on the right track to reach the answer.

Complete step-by-step answer:

Given equation is

$x = 2 + {2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}}$

So, we have to find out the value of ${x^3} - 6{x^2} + 6x$.

Now in given equation take 2 to L.H.S and take cube on both sides we have,

$ \Rightarrow x - 2 = {2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}}$ ………………….. (1)

\[ \Rightarrow {\left( {x - 2} \right)^3} = {\left( {{2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}}} \right)^3}\]

Now as we know ${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}$ and ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}$ so, apply this property in above equation we have,

\[ \Rightarrow {x^3} - {2^3} - 3\left( {{x^2}} \right)\left( 2 \right) + 3x\left( {{2^2}} \right) = {\left( {{2^{\dfrac{2}{3}}}} \right)^3} + {\left( {{2^{\dfrac{1}{3}}}} \right)^3} + 3{\left( {{2^{\dfrac{2}{3}}}} \right)^2}\left( {{2^{\dfrac{1}{3}}}} \right) + 3\left( {{2^{\dfrac{2}{3}}}} \right){\left( {{2^{\dfrac{1}{3}}}} \right)^2}\]

Now simplify the above equation we have,

\[ \Rightarrow {x^3} - 8 - 6{x^2} + 12x = {2^2} + 2 + 3\left( {{2^{\dfrac{2}{3}}}} \right)\left( {{2^{\dfrac{1}{3}}}} \right)\left( {{2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}}} \right)\]

Now from equation (1) we have,

\[ \Rightarrow {x^3} - 8 - 6{x^2} + 12x = {2^2} + 2 + 3\left( {{2^{\dfrac{2}{3} + \dfrac{1}{3}}}} \right)\left( {x - 2} \right)\]

Now simplify the above equation we have,

\[ \Rightarrow {x^3} - 8 - 6{x^2} + 12x = 6 + \left( {3 \times 2\left( {x - 2} \right)} \right)\]

\[ \Rightarrow {x^3} - 8 - 6{x^2} + 12x = 6 + 6x - 12 = 6x - 6\]

\[ \Rightarrow {x^3} - 8 - 6{x^2} + 6x = - 6\]

\[ \Rightarrow {x^3} - 6{x^2} + 6x = 8 - 6 = 2\]

So the required value of ${x^3} - 6{x^2} + 6x$ is 2.

So, this is the required answer.

Note – Whenever we face such types of problems the key concept is simply not to substitute the value of x in the given expression but somehow to simply and to change the expression into a bigger expression containing sub expressions whose values are known to us. This concept will help you get on the right track to reach the answer.

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