Question

If the value of x, $x=2+2^{{\displaystyle \frac{2}{3}}}+2^{{\displaystyle \frac{1}{3}}}$, then the value of $x^3-6x^2+6x$ is 

(a) 3

(b) 2

(c) 1

(d) None of these

Answer 1

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
$x=2+2^{{\displaystyle \frac{2}{3}}}+2^{{\displaystyle \frac{1}{3}}}$
So, we have to find out the value of $x^3-6x^2+6x$.
Now in given equation take 2 to L.H.S and take cube on both sides we have,
$\Rightarrow x-2=2^{{\displaystyle \frac{2}{3}}}+2^{{\displaystyle \frac{1}{3}}}$ ………………….. (1)
$$\Rightarrow {\left(x-2\right)}^3={\left(2^{{\displaystyle \frac{2}{3}}}+2^{{\displaystyle \frac{1}{3}}}\right)}^3$$
Now as we know ${\left(a-b\right)}^3=a^3-b^3-3a^{2}b+3a{b}^2$ and ${\left(a+b\right)}^3=a^3+b^3+3a^{2}b+3a{b}^2$ so, apply this property in above equation we have,
$$\Rightarrow x^3-2^3-3\left(x^2\right)\left(2\right)+3x\left(2^2\right)={\left(2^{{\displaystyle \frac{2}{3}}}\right)}^3+{\left(2^{{\displaystyle \frac{1}{3}}}\right)}^3+3{\left(2^{{\displaystyle \frac{2}{3}}}\right)}^2\left(2^{{\displaystyle \frac{1}{3}}}\right)+3\left(2^{{\displaystyle \frac{2}{3}}}\right){\left(2^{{\displaystyle \frac{1}{3}}}\right)}^2$$
Now simplify the above equation we have,
$$\Rightarrow x^3-8-6x^2+12x=2^2+2+3\left(2^{{\displaystyle \frac{2}{3}}}\right)\left(2^{{\displaystyle \frac{1}{3}}}\right)\left(2^{{\displaystyle \frac{2}{3}}}+2^{{\displaystyle \frac{1}{3}}}\right)$$
Now from equation (1) we have,
$$\Rightarrow x^3-8-6x^2+12x=2^2+2+3\left(2^{{\displaystyle \frac{2}{3}}+{\displaystyle \frac{1}{3}}}\right)\left(x-2\right)$$
Now simplify the above equation we have,
$$\Rightarrow x^3-8-6x^2+12x=6+\left(3\times 2\left(x-2\right)\right)$$
$$\Rightarrow x^3-8-6x^2+12x=6+6x-12=6x-6$$
$$\Rightarrow x^3-8-6x^2+6x=-6$$
$$\Rightarrow x^3-6x^2+6x=8-6=2$$
So the required value of $x^3-6x^2+6x$ 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.