Solve the equation x*x*x....=25

[Math Tutor]

Solve the equation:
[tex]x\sqrt{x\sqrt{x\sqrt{x...}}}=25[/tex].

[Guest]

[tex]x\sqrt{x\sqrt{x\sqrt{x...}}}=x^1x^{\frac{1}{2}}x^{\frac{1}{4}}x^{\frac{1}{8}}...=x^{(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8} ...)}[/tex]

[tex]\sum_{n=1}^{\infty }[/tex] [tex]\frac {1}{n}=2[/tex].... [tex]x^2=25[/tex]
[tex]x=5[/tex] [tex]x =-5[/tex] but [tex]x\ge 0[/tex] so .... [tex]x=5[/tex] one and only solution

[Guest]

Yeah. Nice solution above. Still too long. For this case i put in relation l (the limit) and x:

l(l-x^2)=0 => l = x^2 => x = +/- sqrt(l) => 5 (because x>=0 must hold on)

Q.E.D.

y0urshad0w (YS)

[Guest]

it's not that long ;d
next problem ?