# Root Calculator + Online Solver With Free Steps

The Root Calculator finds the square super-root of a given number, variable(s), or some mathematical expression. The square super-root (denoted as ssrt(x), ssqrt(x), or $\sqrt{x}_s$) is a relatively rare mathematical function.

ssrt(x) represents the inverse operation of tetration (repeated exponentiation), and its calculation involves the Lambert W function or the iterative approach of the Newton-Raphson method. The calculator uses the former method and supports multi-variable expressions.

## What Is the Root Calculator?

The Root Calculator is an online tool that evaluates the square super-root of some input expression. The input value can contain multiple variable terms such as x or y, in which case the function displays a plot of the results over a range of the input values.

The calculator interface consists of a single, descriptive text box labeled “Find the square super-root of,” which is quite self-explanatory – you enter the value or variable term you want to find here, and that is it.

## How To Use the Root Calculator?

You can use the Root Calculator by entering the number whose square super-root is required. You may also enter variables. For example, suppose you want to find the square super-root of 27. That is, your problem looks like this:

$\text{ssqrt}(27) \,\, \text{or} \,\, \text{ssrt}(27) \,\, \text{or} \,\, \sqrt{27}_s$

Then you can use the calculator to solve it in just two steps as follows.

### Step 1

Enter the value or expression to find the square super-root for into the input text box. In the example, this is 27, so enter “27” without quotes.

### Step 2

Press the Submit button to get the results.

### Results

The results are expansive, and which sections show depends on the input. The possible ones are:

1. Input: The input expression in the standard form for square super-root calculation with the Lambert W function: $e^{ W_0(\ln(x)) }$ where x is the input.
2. Result/Decimal Approximation: The square super-root calculation result – could be either a real or complex number. In the case of variable inputs, this section does not show.
3. 2D/3D Plots: The 2D or 3D plots of the result over a range of values for variable terms – replaces the “Result” section. It does not appear when there are more than two variables involved, or nor variables at all.
4. Number Line: The result’s value as it falls onto the number line – does not show if the result is complex.
5. Alternate Forms/Representations: Other possible representations of the square super-root formulation, like the common fraction form: $e^{ W(\ln(x)) } = \frac{\ln(x)}{W(\ln(x))}$ where x is the input.
6. Integral Representations: More alternative representations in the form of integrals if possible.
7. Continued Fraction: The “continued fraction” of the result in the linear or fraction format. It only appears if the result is a real number.
8. Alternate Complex Forms/Polar Form: Exponential Euler, trigonometric, and polar form representations of the result – only shown if the result is a complex number.
9. Position in the Complex Plane: A point visualized at the result coordinates on the complex plane – only appears if the result is a complex number.

## How Does the Root Calculator Work?

The Root Calculator works by using the following equations:

$\text{ssrt}(y) \,\, \text{where} \,\, y = x^x \,\, \vert \,\, x \in +\mathbb{R} \tag*{(1)}$

And its eventual formulation as the exponential of the Lambert W function:

$\text{ssrt}(y) = e^{W(\ln y)} = \frac{\ln y}{W(\ln y)} \tag*{(2)}$

### Tetration and Square Super-Roots

Tetration is the operation of repeated exponentiation. The $n^{th}$ tetration of a number x is denoted by:

${}^{n}x = x \upuparrows n = x^{x^{\cdot^{\cdot^{\cdot^{x}}}}}$

It is convenient to assign a subscript to each instance of x as $x_1,\, x_2,\, x_3,\, \ldots,\, x_n = x$:

${}^{n}x = x_1^{x_2^{\cdot^{\cdot^{\cdot^{x_n}}}}}$

Thus there are n copies of x, repeatedly exponentiated n-1 times. Think of x1 as level 1 (lowest or base), x2 as level 2 (1st exponent), and xn as level n (highest or (n-1)th exponent). Within this context, it is sometimes referred to as a power tower of height n.

The square super-root is the reverse operation of the second tetration $x^x$. That is, if:

$y = x^x \iff \text{ssrt}(y) = \sqrt{y}_s = x$

Solving $y = x^x$ for x (the same process as finding an inverse function) leads to the formulation of the square super-root in equation (2).

### Lambert W Function

In equation (2), W represents the Lambert W function. It is also called the Product Logarithm or Omega function. It is the converse relation of $f(w) = we^w = z$ where w, z $\in \mathbb{C}$, and has the property:

$we^w = z \iff W_k(z) = w \,\, \text{where} \,\, k \in \mathbb{Z}$

It is a multi-valued function with k branches. Only two of these are required when dealing with real numbers, namely $W_0$ and $W_{-1}$. $W_0$ is also called the Principal Branch.

#### Asymptotic Approximation

As tetration involves large values, it is sometimes required to use the asymptotic expansion to estimate the value of the function Wk(x):

\begin{aligned} W_k &= L_1-L_2 + \frac{L_2}{L_1} + \frac{L_2 \!\left(-2+L_2 \right)}{2L_1^2} + \frac{L_2 \!\left( 6-9L_2+2L_2^2 \right)}{6L_1^3} \\ & \quad + \frac{L_2 \!\left(-12+36L_2-22L_2^2+3L_2^3 \right)}{12L_1^4} + \cdots \end{aligned} \tag*{(3)}

Where:

$L_1,\, L_2 = \left\{ \begin{array}{lcl} \ln x,\, \ln (\ln x) & \text{for} & k = 0 \\ \ln(\!-x),\, \ln(\!-\!\ln(\!-x)) & \text{for} & k = -1 \end{array} \right.$

### Number of Solutions

Recall that inverse functions are those that provide a unique, one-to-one solution. The square super-root is not technically an inverse function because it involves the Lambert W function in its calculations, which is a multi-valued function.

Because of this, the square super-root might not have a unique or single solution. Unlike square roots, however, finding the exact number of square super-roots (called the $n^{th}$ roots) is not simple. In general, for ssrt(x), if:

1. x > 1 in ssrt(x), there exists one square super-root also greater than 1.
2. $e^{-\frac{1}{e}}$ = 0.6922 < x < 1, then there are potentially two square super-roots between 0 and 1.
3. 0 < x < $e^{-\frac{1}{e}}$ = 0.6922, the square super-root is complex, and there are infinitely many possible solutions.

Note that in the case of many solutions, the calculator will present one.

## Solved Examples

### Example 1

Find the square super-root of 256. What is the relationship between the result and 256?

### Solution

Let y be the desired result. We then require:

$y = \sqrt{256}_s$

On inspection, we see that this is a simple problem.

$\because 4^4 = 256 \, \Rightarrow \, y = 4$

No need to compute the long way for this!

### Example 2

Evaluate the third tetration of 3. Then, find the result’s square super-root.

### Solution

$3^{3^{3}} = 7.6255 \!\times\! 10^{12}$

Using equation (2), we get:

$\sqrt{7.6255 \!\times\! 10^{12}}_s = e^{ W \left( \ln \left(7.6255 \!\times\! 10^{12} \right) \right) } = \frac{\ln \!\left( 7.6255 \!\times\! 10^{12} \right)}{W \!\left( \ln \!\left( 7.6255 \!\times\! 10^{12} \right) \right)}$

Using the approximation in equation (3) up to three terms, we get:

$\sqrt{7.6255 \!\times\! 10^{12}} \approx \mathbf{11.92}$

Which is close to the calculator’s result of 11.955111.

### Example 3

Consider the function f(x) = 27x. Plot the square super-root for this function over the range x = [0, 1].

### Solution

The calculator plots the following:

Figure 1

All graphs/images were created with GeoGebra.