# 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:

**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.**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.**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.**Number Line:**The result’s value as it falls onto the number line – does not show if the result is complex.**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.**Integral Representations:**More alternative representations in the form of integrals if possible.**Continued Fraction:**The “continued fraction” of the result in the linear or fraction format. It only appears if the result is a real number.**Alternate Complex Forms/Polar Form:**Exponential Euler, trigonometric, and polar form representations of the result – only shown if the result is a complex number.**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:

- x > 1 in ssrt(x), there exists one square super-root also greater than 1.
- $e^{-\frac{1}{e}}$ = 0.6922 < x < 1, then there are potentially two square super-roots between 0 and 1.
- 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:

*All graphs/images were created with GeoGebra.*