This x Is Realllllllly Tucked Away

Calculus Level 1

$\sqrt{\sqrt{\sqrt{\cdots\sqrt{x}}}} = \ ?$

For all positive $x$ , what is the value of the expression above?

Clarification : There are infinitely many radicals.

0 1

3 solutions

Mehul Arora
Feb 12, 2016

$\huge {\sqrt {\sqrt {\sqrt {\cdots \sqrt {x}}}} = x^{( \dfrac {1}{2} \times \dfrac {1}{2} \times \cdots) }}$

$\huge {= x ^ {(\dfrac {1}{2^ {\infty}})} = x^ 0 = 1}$

True because: $2^ {\infty}$ diverges and goes to $\infty$ . Also, $\dfrac {1}{\infty}$ tends to 0.

Chaitnya Shrivastava
Feb 11, 2016

Let the given expression be equal to y Now squaring both sides gives us 》 $y^2$ = y Solving the equation we get y=0 or y=1 but y not equals 0 hence y=1 or the value of above expression is 1

why y not equals 0?

Hải Trung Lê - 5 years, 4 months ago

because √(√2) > √2

Ilan Amity - 5 years, 4 months ago

I'm sorry sir that isn't completely true. Also, that's not the reason for y to be positive.

Mehul Arora - 5 years, 4 months ago

Because $x$ is a positive integer. It does not matter how many times you root it, It will still end up in a positive number.

Mehul Arora - 5 years, 4 months ago

oh, thank you, but are there any logical ways to prove that?

Hải Trung Lê - 5 years, 4 months ago

@Hải Trung Lê – Logical as in? I mean, how exactly do you define a "logical way"?

Mehul Arora - 5 years, 4 months ago

Tanishq Bhatia
Feb 12, 2016

Try putting more options.The question asks for A positive value and the options contain only one positive value.

It only says $x$ is positive, not the answer.

Jack Rawlin - 5 years, 4 months ago