Please help me understand how this works: sqrt[sqrt[sqrt[sqrt[2]]]]=2^x

[marsh]

The solution is right there too but I just need to understand what is happening

 

[Deleted member 4993]

The solution is right there too but I just need to understand what is happening

Exactly which step you don't understand?

The first step was to re-write the inner-most √2 as 2^½

and continue re-writing from there!

 

[HallsofIvy]

Here's one method: From \(\displaystyle \sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}= 2^x\), square both sides: \(\displaystyle 2\sqrt{2\sqrt{2\sqrt{2}}}= 2^{2x}\). Divide both sides by 2 so \(\displaystyle \sqrt{2\sqrt{2\sqrt{2}}}= 2^{2x- 1}\).

Now square again: \(\displaystyle \sqrt{2\sqrt{2}}= 2^{2(2x- 1)}= 2^{4x- 2}\).

Again, divide by 2 to get \(\displaystyle \sqrt{2\sqrt{2}}= 2^{4x- 3}\).

Square a third time: \(\displaystyle 2\sqrt{2}= 2^{8x- 6}\).

Again, divide by 2: \(\displaystyle \sqrt{2}= 2^{8x- 7}\)

Square a fourth time: \(\displaystyle 2= 2^1= 2^{16x- 14}\) so we must have 16x- 14= 1.

Another method. Use the facts that \(\displaystyle \sqrt{a}= a^{1/2}\), that \(\displaystyle (a^p)^q= a^{pq}\) and \(\displaystyle (a^p)(a^q)= a^{p+q}\).

So \(\displaystyle 2\sqrt{2}= 2^1(2^{1/2})= 2^{3/2}\)
\(\displaystyle \sqrt{2\sqrt{2}}= (2^{3/2})^{1/2}= 2^{3/4}\)
\(\displaystyle 2\sqrt{2\sqrt{2}}= 2^1(2^{3/4})= 2^{7/4}\)
\(\displaystyle \sqrt{2\sqrt{2\sqrt{2}}}= 2^{7/8}\)
\(\displaystyle 2\sqrt{2\sqrt{2\sqrt{2}}}= 2^{15/8}\)

Finally, \(\displaystyle \sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}= 2^{15/16}\).