If $\sqrt{\sqrt{\sqrt{5x}}}=\sqrt[6]{4x},$ what is the range of the possible values of $x?$

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Gmat_mission: Legendary Member; Posts: 1622; Joined: Thu Mar 01, 2018 7:22 am; Followed by:2 members

If $\sqrt{\sqrt{\sqrt{5x}}}=\sqrt[6]{4x},$ what is the range of the possible values of $x?$

by Gmat_mission » Sun Mar 14, 2021 7:36 am

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If $\sqrt{\sqrt{\sqrt{5x}}}=\sqrt[6]{4x},$ what is the range of the possible values of $x?$

A. $0$

B. $14$

C. $\dfrac{125}{256}$

D. $\dfrac12$

E. $\dfrac{131}{256}$

Answer: C

Source: GMAT Club Tests

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Source: Beat The GMAT — Problem Solving | Sun Mar 14, 2021 7:36 am

deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pm; Followed by:5 members

Re: If $\sqrt{\sqrt{\sqrt{5x}}}=\sqrt[6]{4x},$ what is the range of the possible values of $x?$

by deloitte247 » Wed Mar 17, 2021 12:33 am

What is the range of the possible value of x?
$$\left(5x\right)^{\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}}=\left(4x\right)^{\frac{1}{6}}$$
$$\left(5x\right)^{\frac{1}{8}}=\left(4x\right)^{\frac{1}{6}}$$
$$5x=\left[\left(4x\right)^{\frac{1}{6}}\right]^8$$
$$5x=\left(4x\right)^{\frac{8}{6}}$$
$$5x=\left(4x\right)^{\frac{4}{3}}$$
$$5x=4^{\frac{4}{3}}\cdot x^{\frac{4}{3}}$$
$$\frac{5x}{x\cdot4^{\frac{4}{3}}}=\frac{\left(4^{\frac{4}{3}}\cdot x^{\frac{4}{3}}\right)}{x\cdot4^{\frac{4}{3}}}$$
$$\frac{5}{4^{\frac{4}{3}}}=\frac{x^{\frac{4}{3}}}{x^1}$$ $$\frac{5}{4^{\frac{4}{3}}}=x^{\frac{4}{3}-1}$$ $$\frac{5}{4^{\frac{4}{3}}}=x^{\frac{4-3}{3}}$$
$$x^{\frac{1}{3}}=\frac{5}{4^{\frac{4}{3}}}$$
$$\left(x^{\frac{1}{3}}\right)^3=\left(\frac{5}{4^{\frac{4}{3}}}\right)^3$$
$$x=\frac{5^3}{4^{\frac{4}{3}\cdot3}}=\frac{5^3}{4^4}=\frac{125}{256}$$

Answer = C