Question

If √(x + √(x + √(x + ...))) = √(x√(x√(x...))) then find the value of x.​

Answer 1

Answer:

x = 0 or x = 2

Step-by-step explanation:

Let say

√(x + √(x + √(x + ...))) = √(x√(x√(x...)))  = K

√(x + √(x + √(x + ...)))  = K

Squaring both sides

=> x + √(x + √(x + √(x + ...))) = K²

=> x + K = K²    Eq 1

√(x√(x√(x...)))  = K

Squaring both sides

=> x√(x√(x√(x...)))  = K²

=> xK = K²

=> x = K

putting this value in eq 1

x + x = x²

=> 2x = x²

=> x² - 2x = 0

=> x(x - 2) = 0

=> x = 0 or x = 2

√(x + √(x + √(x + ...))) = √(x√(x√(x...))) = 2  or 0

Answer 2

Step-by-step explanation:

Let

$let \: \sqrt{x \sqrt{x \sqrt{x \sqrt{x......} } } } \: equal \: to \: y$

Then y^2 = xy

$let \: \sqrt{x + \sqrt{x + \sqrt{x} } } \: also \: equal \: to \: y \: as \: it \: is \: given \: that \: both \: are \: equal$

Then y^2 = x + y

if y^2 = xy, then x = y

So x^2 = 2x

x = 2

I hope iam right...

Apologize if not..