Simple but Nice #2

$X=2+{ 2 }^{ \frac { 2 }{ 3 } }+{ 2 }^{ \frac { 1 }{ 3 } }\\ X={ 2 }^{ \frac { 1 }{ 3 } }({ 2 }^{ \frac { 2 }{ 3 } }+{ 2 }^{ \frac { 1 }{ 3 } }+1)\\ X={ 2 }^{ \frac { 1 }{ 3 } }({ 2 }^{ \frac { 2 }{ 3 } }+{ 2 }^{ \frac { 1 }{ 3 } }+2-2+1)\\ X={ 2 }^{ \frac { 1 }{ 3 } }(X-1)\\ Cubing\quad both\quad sides\\ { X }^{ 3 }=2({ X }^{ 3 }-1-3{ X }^{ 2 }+3X)\\ { X }^{ 3 }=2{ X }^{ 3 }-2-6{ X }^{ 2 }+6X\\ { X }^{ 3 }-6{ X }^{ 2 }+6X=\boxed{2}$

$x-2=2^{\frac{1}{3}}( 2^{\frac{1}{3}}+1)$ Cubing gives us, $x^3-6x^2+12x-8=6+6× 2^{\frac{1}{3}}( 2^{\frac{1}{3}}+1)$ $=6+6(x-2)$ $\Rightarrow x^3-6x^2+6x=\boxed{2}$

We are given that $x=2+\sqrt[3]{2}+\sqrt[3]{4}$ . Let $x=y+2$ , hence $y=\sqrt[3]{2}+\sqrt[3]{4}$ .

Cube both sides, simplify and substitute again the value of $y$ :

$y^3=(\sqrt[3]{2}+\sqrt[3]{4})^3 \\y^3=2+3\sqrt[3]{8}(\sqrt[3]{2}+\sqrt[3]{4})+4 \\ y^3=6y+6 \\ y^3-6y-6=0$

Finally, since $x=y+2$ , we know that $y=x-2$ , so:

$(x-2)^3-6(x-2)-6=0 \\ x^3-6x^2+12x-8-6x+12-6=0 \\ x^3-6x^2+6x=\boxed{2}$

$x=2+2^{2/3}+2^{1/3}$

$x-2=2^{2/3}+2^{1/3}$

Cubing both sides:

$(x-2)^3=6+6 \times 2^{2/3}+6 \times 2^{1/3} = 6x - 6$

$(x-2)^3 - 6x + 8 = 2$

$\boxed { x^3 - 6x^2 + 6x =2 }$

: x - 1 = 2^(-1/3)*x So (x-1)^3 = 0.5x^3 x^3 - 3x^2 + 3x - 1 = 0.5x^3 i.e. x^3 - 6x^2 + 6x = 2

Doing it the long way. $x^2=2^2+(2^{\frac {1} {3}})^2+(2^{\frac {2} {3}})^2+2(2)(2^{\frac {2} {3}})+2(2)(2^{\frac {1} {3}})+2(2^{\frac {1} {3}})(2^{\frac {2} {3}})$ $x^2=4+2^{\frac {2} {3}}+2^{\frac {4} {3}}+ 4(2^{\frac {2} {3}})+ 4(2^{\frac {1} {3}})+2(2)$ $x^2= 8+ 5(2^{\frac {2} {3}})+ 4(2^{\frac {1} {3}})+2^{\frac {4} {3}}$ $x^2= 8+ 5(2^{\frac {2} {3}})+ 4(2^{\frac {1} {3}})+2(2^{\frac {1} {3}})$ $x^2= 8+ 5(2^{\frac {2} {3}})+ 6(2^{\frac {1} {3}})$ $x^3=x^2 \times x$ $x^3= [8+ 5(2^{\frac {2} {3}})+ 6(2^{\frac {1} {3}})] \times [2+2^{\frac {2} {3}}+2^{\frac {1} {3}})]$ $x^3=38+ 24(2^{\frac {2} {3}})+ 30(2^{\frac {1} {3}})$

Thus $x^3-6x^2+6x=2$

Using a calculator: I can approximate x quickly as 2 + 1.587 + 1.26 = 4.8476 Plugging into x^3-6x^2+6x I get 113.914 - 140.995 + 29.0856 = 2.0046. I gave 2 as my answer. If it would have been wrong, then I would have gone the other route, performing the algebraic cube.