# Rational Exponents Calculator + Online Solver With Free Steps

The **Rational Exponents Calculator** evaluates the exponent of a given input number or expression, provided the exponent is rational.

Exponents, indicated by ‘^’ or superscript as in $x^n$ with n as the exponent, depict the operation of “raising to a power.” In other words, this means multiplying the expression or number by itself n times:

\[ y^n = y \quad \underbrace{\times}_{k\,=\,1} \quad y \quad \underbrace{\times}_{k\,=\,2} \quad \cdots \quad \underbrace{\times}_{k\,=\,n-1} \quad y \quad \underbrace{\times}_{k\,=\,n} \quad y \]

Which shortens to:

\[ y^n = \prod_{k=1}^n y \]

The calculator supports **variable** **and multi-variable inputs** for both the expression and the exponent. The result sections change quite a lot depending on both the type and magnitude of the input. Thus, the calculator always presents the results in the most relevant and appropriate form.

## What Is the Rational Exponents Calculator?

**The Rational Exponents Calculator is an online tool that raises an input number or expression (with or without variables) to the power of a provided rational exponent. The exponent may also be variable. **

The **calculator interface** consists of two text boxes placed next to each other, separated by a **‘^’** indicating the exponentiation. In the first text box to the left of the ^ symbol, you enter the number or expression whose exponent you want to evaluate. In the second box to the right, you enter the value of the exponent itself.

## How To Use the Rational Exponents Calculator?

You can use the **Rational Exponents Calculator** to find the exponent of a number or an expression by entering the number/expression and the value of the exponent into the text boxes.

For example, suppose you want to evaluate $37^4$. You can use the calculator to do so using the step-by-step guidelines below.

### Step 1

Enter the number/expression in the first text box to the left. For the example, enter “37” without quotes.

### Step 2

Enter the exponent value in the second text box to the right. For the example, you would enter “4” without quotes here.

### Step 3

Press the **Submit** button to get the results.

### Results

The result section is expansive and depends heavily on the type and magnitude of the input. Two of these sections, however, are always displayed:

**Input:**The input expression as the calculator interprets it in LaTeX format (for manual verification). For our example, 37^4.**Result:**The actual result value. For our example, this is 1874161.

Let a, b be two constant coefficients, and x, y be two variables for the following text.

#### Constant Value to a Constant Exponent

Our example falls in this category. The results contain (sections marked with * appear always):

***Number Line:**The number as it falls onto the number line (up to an appropriate zoom level).

**Number Name:**The pronunciation of the resulting value – is only displayed if the result is in non-scientific notation.

**Number Length:**The number of digits in the result – only appears when it exceeds five digits. For our example, this is 7.

**Visual Representation:**The resulting value in the form of dots. This section shows only when the result is an integer value strictly smaller than 39.**Comparison:**This section shows if the resulting value compares to some known quantity. For our example, it is almost half of the possible arrangements for a 2x2x2 Rubik’s cube ($\approx$ 3.7×10^6).

Other sections might appear as well for decimal exponents.

#### Variable Value to a Constant Exponent

For input expressions of the type $f(x) = x^a$ or $f(x,\, y) = (xy)^a$, the following sections appear:

**2D/3D Plot:**Plot of the function over a range of the variable’s values. 2D if only one variable is present, 3D if two, and none if more than two.

**Contour Plot:**The contour plot for the resulting expression – only appears if there is a 3D plot for the result.

**Roots:**The roots of the expression, if they exist.

**Polynomial Discriminant:**The discriminant of the resulting expression. Found using the known equations for low-degree polynomials.

**Properties as a Function:**The domain, range, parity (even/odd function), and periodicity (if it exists) for the resulting expression expressed as a function.

**Total/Partial Derivatives:**The total derivative of the resulting expression if only one variable is present. Otherwise, for more than one variable, these are partial derivatives.

**Indefinite Integral:**The indefinite integral of the resulting function w.r.t one variable. If more than one variable is present, the calculator evaluates the integral w.r.t. the first variable in alphabetical order.

**Global Minima:**The minimum value of the function – only appears when roots exist.

**Global Maxima:**The maximum value of the function – only shows if roots exist.

**Limit:**If the resulting expression represents a converging function, this section shows the convergence value as a limit of the function.

**Series Expansion:**The result expanded about a value of the variable using a series (generally Taylor). If more than one variable, the expansion is done w.r.t. the first variable in alphabetical order.

**Series Representation:**The result in the form of a series/summation – shown only if possible.

#### Constant Value to a Variable Exponent

For input expressions of the type $a^x$ or $a^{xy}$, the results contain the same sections as in the previous case.

#### Variable Value to a Variable Exponent

For input expressions of the type $(ax)^{by}$, the calculator again shows the same sections as in the previous variable cases.

## Solved Examples

### Example 1

Evaluate the expression $\ln^2(40)$.

### Solution

Given that:

\[ \ln^2(40) = (\ln40)^2 \]

**ln 40 = 3.68888 **

\[ \Rightarrow \, \ln^2(40) = (3.68888)^2 = \left( \frac{368888}{100000} \right)^2 = \mathbf{13.60783} \]

### Example 2

Plot the function $f(x, y) = (xy)^2$.

### Solution

Given that:

\[ (xy)^2 = x^2y^2 \]

The calculator plots the function as below:

And the contours:

### Example 3

Evaluate:

\[ 32^{2.50} \]

### Solution

The exponent 2.50 can be expressed as the improper fraction 250/100 and simplified to 5/2.

\[ \therefore \, 32^{2.50} = 32^{ \frac{5}{2} } = \left( 32^\frac{1}{2} \right)^5 \]

\[ 32^{2.50} = \left( \sqrt[2]{32} \right)^5 = \left( \sqrt[2]{2^4 \cdot 2} \right)^5 \]

\[ \Rightarrow 32^{2.50} = (4 \sqrt[2]{2})^5 = (4 \times 1.41421)^5 = \mathbf{5792.545794} \]

*All graphs/images were created with GeoGebra.*