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# Exponents Calculator + Online Solver With Free Steps

The **Exponents Calculator** is used to compute the exponent function of a number. It takes the number and the exponent of the number as input and outputs the **multiplication** result.

An exponent is also referred to as the **power** or **degree** of a number. The Exponents Calculator multiplies the same number many times according to the exponent.

The number multiplying several times is known as the “**base**”. The exponent is written as a **superscript** to the base. The exponent defines how often the base number needs to be multiplied to get the final result.

Suppose the** base** number is 2 and the **exponent** number is 3. The exponent 3 will be written in superscript to the base number 2. It will be read as “**2 raised to the power 3**” and written as $2^3$. It means that the number 2 should be **multiplied by itself** 3 times to get the final result. The result will be 2×2×2 which is 3.

To **generalize** the exponent function, suppose b is multiplied by itself m times. It will be written as $b^m$ where **b** and **m** are both integers.

**Exponents** can also be **negative** numbers. Suppose the base number is 5 and the exponent number is -4. It will be written as $5^{-4}$. By multiplying and dividing by $5^4$, we will achieve:

\[ 5^{-4} = 5^{-4} × \frac{5^4}{5^4} \]

If the bases are the same and are being multiplied, the exponents **add** up as:

\[ \frac{ 5^{-4+4} }{ 5^4 } = \frac{ 5^{0} }{ 5^4 } \]

Any non-zero number raised to the power zero is **one**. So, the result is $\dfrac{ 1 }{ 5^4 }$.

To generalize this result, if **a** is multiplied **-n** times provided that a is not equal to zero then:

\[ a^{-n} = \frac{1}{ a^{n} } \]

The Calculator also takes in negative exponents to compute the multiplication. The **square root** is a special exponent function with the exponent as **1/2**. The **cube root** refers to the exponent **1/3**.

## What Is an Exponents Calculator?

**The Exponents Calculator is an online tool that is used to determine the multiplication of a number by using the exponent function. The base and the exponent are the inputs of the Exponents Calculator.**

The base and the exponent can be a positive number, a negative number, or a fraction.

If the output contains a **decimal**, the calculator shows the decimal approximation of the number. It also shows the **continued fraction** and the real and imaginary roots of the output in polar form.

The **graph plot** for all the roots of the resultant number is also displayed by the calculator.

If the base and exponent entered by the user are **variables**, the calculator also shows the 3D plot, contour plot, periodicity, derivative, indefinite integral, and the limit for the entered input.

## How To Use the Exponents Calculator

The user can use the Exponents Calculator by following the steps given below.

### Step 1

The user must first enter the **base** number in the input window of the calculator. It should be entered in the block before the symbol “ ^ ”.

The base number is the number that needs to be multiplied as many times as specified by the exponent number.

The calculator uses the base number **5** for the **default** example.

### Step 2

The user must now enter the** exponent** number in the calculator’s input window. It should be entered in the block after the symbol “ ^ ”.

The exponent is the **power** and denotes how many times the base number needs to be multiplied by itself to get the final result.

The exponent can be a** rational **number and an **integer** depending upon the user. If the exponent is zero, the result will always be one.

For the **default** example, the exponent used is **2** which denotes the square of a number.

### Step 3

The user must now press the “**Submit**” button for the calculator to process the base and the exponent. It calculates the result as given below.

### Output

The Exponents Calculator computes the Output in the five windows given below.

#### Input

This window shows the **input interpretation** of the calculator. It shows the base and the exponent as entered by the user in the input window.

For the **default** example, the calculator displays the Input as follows:

\[ \text{Input} = 5^2 \]

#### Result

The Calculator computes the **multiplication** of the base number by using the exponent function and displays the result in this window.

The user can press “Need a step-by-step solution to this problem?” for all the **mathematical steps** required to solve the particular problem.

For the **default** example, the base is 5 and the exponent is 2. The calculator computes 5 × 5 and shows the final result to be **25**.

#### Number Line

The Number Line window shows the final result on the **number line**. It is represented by a **dot** on the number line. The number line is a horizontal line with the numbers placed at regular **intervals** in ascending order.

The calculator shows the result **25** for the **default** example on the number line as in figure 1.

#### Number Name

The calculator displays the **name** of the resulting number in this window. It shows the number in words. For the **default** example, it displays the number name as **twenty-five**.

#### Visual Representation

The calculator also displays the visual representation of the result in this output window. The visual representation shows the **number of dots** according to the result value.

The calculator displays twenty-five dots in the Visual Representation window for the default example.

## Solved Examples

The following examples are solved through the Exponents Calculator.

### Example 1

Calculate the result for the base fraction as 1/4 and the exponent as -3.

### Solution

The user must first enter the **base** 1/4 and the **exponent** 3 as specified in the example. The base should be entered in **round brackets** for the calculator to assume the power -3 on the complete fraction and not only on 4.

After submitting the input values, the calculator computes the **Output** and displays it under multiple headings.

At first, the calculator interprets the **input** and shows it as given below.

\[ \text{Input} = \frac{ 1 }{ { ( \frac{1}{4} ) }^3 } \]

The calculator computes the exponent function and displays the **Exact Result** as **64**. It shows this result on the number line as shown in figure 2.

The calculator also displays the Number Name of the result value as **sixty-four**.

### Example 2

Calculate 6×6×6×6×6 by using the exponent function.

### Solution

The user must first identify the base and the exponent to input into the calculator. The **base** is **6** as it is the number being multiplied. The **exponent** is **5** as the number 6 is multiplied 5 times by itself.

The base number 6 and the exponent 5 should be entered in the **input** tab of the calculator. After submitting the result, the calculator computes the **output** as given below.

The **Input** Interpretation shows the input base and exponent as entered by the user. The Calculator displays it as follows:

\[ \text{Input} = 6^5 \]

The calculator computes the multiplication and displays the **final answer** to be **7776**. It also shows this result on the number line as in figure 3.

The calculator shows the resulting number in words as **seven thousand, seven hundred and seventy-six**.

### Example 3

Calculate the result if the base number is 72 and the exponent is 1/2.

### Solution

The user must first enter the **base** number and the **exponent** in the input window of the calculator. After pressing the “**Submit**” button, the calculator displays the output in multiple windows.

The **Input** window shows the input interpretation by the calculator. For this example, it shows the Input as follows:

\[ \text{Input} = \sqrt{72} \]

The calculator solves for the base and exponent and outputs the **Result** as:

\[ \text{Result} = 6 \sqrt{2} \]

The **decimal approximation** for the above result shown by the calculator is 8.48528137423857 and so on.

The calculator displays the result on the **number line** as shown in figure 4.

The calculator also shows the Continued Fraction of the result as follows:

\[ \text{ Continued Fraction } = [ 8; \bar{ 2, 16} ] \]

The **Continued fraction** is a fraction whose denominator is a variable plus a fraction and so on. It is a fraction of infinite length.

The calculator also displays all the** second roots** of 72. They can be displayed in polar form, trigonometric form, or radical form. The calculator displays these options on the right-hand side of the window.

The second roots in **polar form** for the result are:

\[ \text{ Real, Principal Root } = 6 \sqrt{2} e^0 ≈ 8.485 \]

\[ \text{ Real Root } = 6 \sqrt{2} e^{ ίπ } ≈ -8.485 \]

The Exponents Calculator also displays the **plot** for all the roots in the **complex plane** for this example. It is shown in figure 5.

*All the images are created using Geogebra.*