Question

if x=2+2power 2/3+2 power 1/3 then find the value of x power 3-6x square +6x​

Answer 1

Answer:

${x}^{3} - 6 {x}^{2} + 6x = 14$

Step-by-step explanation:

Given :

$x \: = \: 2 + {2}^{ \frac{2}{3} } + {2}^{ \frac{1}{3} }$

To find :

${x}^{3} - 6 {x}^{2} + 6x$

Solution :

$x - 2 = {2}^{ \frac{2}{3} } + {2}^{ \frac{1}{3} } \\ cubing \: on \: both \: the \: sides \\ {(x - 2)}^{3} = {( {2}^{ \frac{2}{3} } + {2}^{ \frac{1}{3} }) }^{3} \\ {x}^{3} - 8 - 6x(x - 2) = \\ {( {2}^{ \frac{2}{3} }) }^{3} + {( {2}^{ \frac{1}{3} }) }^{3} + \\ 3 \times {2}^{ \frac{2}{3} } \times {2}^{ \frac{1}{3} } ( {2}^{ \frac{2}{3} } + {2}^{ \frac{1}{3} } )$

$=> {x}^{3} - 6 {x}^{2} + 12x - 8 \: = \\ {2}^{2} + 2 + 3 \times 2( {2}^{ \frac{2}{3} } + {2}^{ \frac{1}{3} } ) \\ 6 \: + 6x \\ (since \: \: {2}^{ \frac{2}{3} } + {2}^{ \frac{1}{3} } = x)$

$=> {x}^{3} - 6 {x}^{2} + 12x - 8 = 6 + 6x \\ => {x}^{3} - 6 {x}^{2} + 6x = 6 + 8 \\ = 14$

Hence, the answer.

Answer 2

Step-by-step explanation:

$\bold{Given : x = 2 + 2^\frac{2}{3} + 2^\frac{1}{3}}$

$\implies(x - 2) = 2^\frac{2}{3} + 2^\frac{1}{3}$

$\bold{Cubing\;on\;both\;sides,\;We\;get\;:}$

$\implies(x - 2)^3 = (2^\frac{2}{3} + 2^\frac{1}{3})^3$

$x^3 + (-2)^3 + 3(x)(-2)^2 + 3(x)^2(-2) = (2^\frac{2}{3})^3 + (2^\frac{1}{3})^3 + 3(2^\frac{2}{3})(2^\frac{1}{3})^2 + 3(2^\frac{2}{3})^2(2^\frac{1}{3})$

$\implies x^3 - 8 + 3(x)(4) - 6x^2 = 2^2 + 2 + 3(2^\frac{2}{3})(2^\frac{2}{3}) + 3(2^\frac{4}{3})(2^\frac{1}{3})$

$\implies x^3 - 6x^2 + 12x - 8 = 4 + 2 + 3(2^\frac{4}{3}) + 3(2^\frac{5}{3})$

$\implies x^3 - 6x^2 + 6x = 8 + 6 + 3(2^\frac{4}{3}) + 3(2^\frac{5}{3}) - 6x$

$\bold{But,\;We\;know\;that\;: x = 2 + 2^\frac{2}{3} + 2^\frac{1}{3}}$

$\implies x^3 - 6x^2 + 6x = 14 + 3(2^\frac{4}{3}) + 3(2^\frac{5}{3}) - 6(2 + 2^\frac{2}{3} + 2^\frac{1}{3})$

$\implies x^3 - 6x^2 + 6x = 14 + 3(2^\frac{4}{3}) + 3(2^\frac{5}{3}) - 12 - 6(2^\frac{2}{3}) - 6(2^\frac{1}{3})$

$\implies x^3 - 6x^2 + 6x = 2 + 3(2^\frac{4}{3}) + 3(2^\frac{5}{3}) - 3(2^(^\frac{2}{3}^+^1^)) - 3(2^(^\frac{1}{3}^+^1^))$

$\implies x^3 - 6x^2 + 6x = 2 + 3(2^\frac{4}{3}) + 3(2^\frac{5}{3}) - 3(2^\frac{4}{3}) - 3(2^\frac{5}{3})$

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