How to factorize the expression $$\sqrt{1+\sqrt{x^2+\sqrt{x^4+\sqrt{x^6}}}}\quad?$$
How to factorize the above expression? I am trying the above question by taking the above expression as $y$ and then squaring both sides. After squaring I am getting $y^2=1+\sqrt{x^2+\sqrt{x^4+\sqrt{ x^6}}}$. Now I shifted $1$ on the left hand side and then again squared. So, again after squaring I am getting $y^4-2y^2+1=x^2 +\sqrt{x^4+\sqrt{x^6}}$. But how to approach further. Kindly help me out with this question.
Too long for a comment.
Deb Sankar Roy, what do you mean by “factorize” ?
$$\sqrt{1+\sqrt{x^2+\sqrt{x^4+\sqrt{x^6}}}}=$$
$$=\sqrt{1+\sqrt{x^2+\sqrt{x^4+\left|x^3\right|}}}=$$
$$=\sqrt{1+\sqrt{x^2+\sqrt{x^2\left(x^2+\left|x\right|\right)}}}=$$
$$=\sqrt{1+\sqrt{x^2+\left|x\right|\sqrt{x^2+\left|x\right|}}}=$$
$$=\sqrt{1+\sqrt{x^2+\left|x\right|\sqrt{\left|x\right|\big(\left|x\right|+1\big)}}}=$$
$$=\sqrt{1+\sqrt{\left|x\right|\left[\left|x\right|+\sqrt{\left|x\right|\big(\left|x\right|+1\big)}\,\right]}}\;.$$
Tell me if it is what you meant.