Super Root Calculator

Find the base that produces a specified power tower result. Enter a target number and the height of the tetration tower to calculate its super root.

Calculate a Super Root

Target Value (x)

The result that the power tower must produce.

Tower Height (n)

Number of repeated exponentiation levels.

Tolerance

Controls the numerical accuracy of the solution.

Maximum Iterations

Maximum number of bisection steps.

What Is a Super Root?

A super root is the base of a power tower when the tower height and final result are already known. It is closely related to tetration, which is repeated exponentiation.

Ordinary roots reverse repeated multiplication. A super root reverses a repeated exponentiation process.

T₁ = a
T₂ = a^a
T₃ = a^{a^a}
T_n = a^T_n-1

Given a target x and tower height n, the calculator solves:

T_n(a) = x

The unknown variable is the base a. The calculator finds a numerical value for that base.

Example: Find the Super Root of 16

Suppose the target is 16 and the tower height is 3. We need to find a base a satisfying:

a^{a^a} = 16

Trying a base of 2 gives:

T₁ = 2
T₂ = 2² = 4
T₃ = 2⁴ = 16

Therefore, the super root is exactly 2 for this example.

How the Super Root Algorithm Works

The calculator receives the target value and tower height.
It starts with a base interval whose lower endpoint is 1.
It evaluates the power tower at the midpoint of the interval.
If the tower result is below the target, it increases the lower bound.
If the tower result is above the target, it decreases the upper bound.
It repeats until the result is sufficiently close to the target or the iteration limit is reached.

This numerical bisection method is suitable for the increasing real-valued tower branch with base at least 1. Very large towers may exceed the floating-point range supported by PHP.

Frequently Asked Questions

1. What does a super root calculate?

It calculates the base that produces a specified result after repeated exponentiation to a specified tower height.

2. Is a super root the inverse of tetration?

It is an inverse problem for tetration with respect to the base. The tower height and final value are known, and the unknown base is calculated.

3. How is a super root different from a super logarithm?

A super root finds the base when the height and target are known. A super logarithm finds the height when the base and target are known.

4. Why does the calculator use bisection?

Bisection repeatedly narrows a range containing the solution. It is a reliable numerical method for the supported increasing power-tower branch.

5. Can every target have a super root?

Under this calculator's convention, the base is restricted to real values at least 1 and the target must be at least 1. Other real or complex branches may require different mathematical definitions and numerical methods.

6. Does tower height affect the answer?

Yes. The same target can have different bases at different tower heights because each additional level applies another exponentiation.