Question

$\sqrt{4},\sqrt[4]{4},\sqrt[8]{4},\sqrt[12]{4},\sqrt[16]{4},...$ are the roots of the equation

• A. $x^2–4=0$
• B. $x^2–4x+6=0$
• C. $x^2–5x+4=0$
• D. $x^3–3x+2=0$

Answer 1

The correct option is C

$x^2–5x+4=0$

Explanation for the correct answer:

Finding the equation using the roots:

It is given that
$\sqrt{4},\sqrt[4]{4},\sqrt[8]{4},\sqrt[12]{4},\sqrt[16]{4},...$are the roots of the derived equation.

$$\begin{gathered} x^2–5x+4+4^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...}=0\left[\because \mathrm{the}\mathrm{value}\mathrm{of}{4}^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...}\mathrm{will}\mathrm{be}\mathrm{small}\mathrm{so}\mathrm{we}\mathrm{neglect}\mathrm{it}\right] \\ {x}^2–5x+4=0 \end{gathered}$$

Therefore, option (C) is the correct answer.