# 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

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

Figure 1

### 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:

Figure 2

And the contours:

Figure 3

### 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.

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

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

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

Figure 4

All graphs/images were created with GeoGebra.