## Exponents Calculator

## Definition – What is an exponent?

An exponent is the number of times to multiply a number by itself.

Write an exponent as a raised number. In the number 2^{4} (2 to the exponent 4, or 2 to the power of 4), the ‘4’ is the exponent. The ‘2’ is the number to multiply by itself 4 times over. In this case 2 x 2 x 2 x 2 = 16.

## Formula – How to Solve for an Exponent

Find exponent of a number by multiplying that number by itself the number of the exponent times.

number^{2} = number x number

number^{3} = number x number x number

number^{4} = number x number x number x number

### Example

3^{2} = 3 x 3 = 9

9^{5} = 9 x 9 x 9 x 9 x 9 = 59,049

5^{10} = 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 9,765,625

## Exponent Rules (Laws of Exponents)

##### Product with same base

To multiply similar bases, keep the base the same and add the exponents.

x^{a} • x^{b} = x^{(a + b)}

Example: 7^{3} • 7^{5} = 7^{(3 + 5)} = 7^{8} = 5,764,801

##### Exponent of an Exponent (or Power to a Power)

To calculate an exponent of an exponent, multiply the exponents together.

(x^{a})^{b }= x^{(a • b)} = x^{ab}

Example: (4^{3})^{2} = 4^{(3 • 2)} = 4^{6} = 4,096

##### Dividing Numbers with Exponents (Quotients with the Same Base)

To divide two bases with the same exponent, subtract the denominator exponent from the numerator exponent.

x^{a} ÷ x^{b} = x^{(a – b )}

Example: 5^{7} ÷ 5^{3} = 5^{(7 – 3)} = 5^{4} = 625

##### Multiplied Numbers to an Exponent

Multiplied numbers to a power can be both raised to that power.

(xy)^{z} = x^{z} • y^{z}

Example: (9x)^{5} = 9^{5}x^{5} = 59,049x^{5}

##### Dividing to an exponent

To divide a fraction that is raised to an exponent, give the exponent to both the numerator and the denominator.

(x ÷ y)^{z} = x^{z} ÷ y^{z}

Example: (7 ÷ 5)^{4} = 7^{4} ÷ 5^{4} = 2,401 ÷ 625 = 3.8416

##### Exponent of 0

Any number to the power of 0 is 1.

x^{0} = 1

Example: 450^{0} = 1

##### Negative Exponents

Negative exponents can be converted to 1 divided by the base^{exponent}

x^{-a} = 1 ÷ 1^{a}

Example: 6^{-4} = 1 ÷ 6^{4} = 1 ÷1,296 = 0.0007716

##### Dividing with a Negative Exponent

Numbers with negative exponents as the denominator can be changed to the numerator and the exponent made positive.

1 ÷x^{-a} = x^{a}

Example: 1 ÷ 3^{-4} = 3^{4} = 81

## How to type exponents

- In Microsoft Word and other Office products, right-click and select Font to open the font menu. Select “Superscript.”
- In Google Docs and other products, hi-light the text you want to have as an exponent. Select “Format” -> “Text” -> “Superscript.”
- In HTML and web documents, surround the exponent with <sup></sup> tags.
- In LaTeX notation, use the ^ symbol before the exponent. If there is more than one character in the exponent, wrap the characters in (brackets).

## Exponents Table

## Frequently Asked Questions

##### What is an exponent (in math)?

An exponent is the number of times to multiply a number by itself. For example, 3 to the 4th (written 3) means 3 x 3 x 3 x 3 = 81. It is not the same as 3 x 4 (12).

##### What is the difference between "Power Of" and "Exponent"?

They are the same thing. Most people use the terms “to the power of” and “to the exponent” interchangeably.

We find that if describing a thing, “the exponent” is a more natural term. (“What is the exponent on the number in that equation?” sounds better than “What is the power of on the number in that equation?”).

If describing the action, “power of” is a more natural term (“Calculate five to the power of three” sounds better than “Calculate five to the exponent three”).

##### What is a negative exponent?

A negative exponent means the number of times to divide the number. 3^{4} (a positive exponent) means to multiply 3 times itself 4 times (3 x 3 x 3 x 3 = 81). 3^{-4} (a negative exponent) means to divide 3 by itself 4 times (3 ÷3 ÷ 3 ÷ 3 = 0.012346).

## Sources and more resources

- A simple introduction to powers and exponents from Math Planet, Purple Math, Math is Fun, and the University of Minnesota School of Public Health.
- Exponent Rules and the Laws of Exponents from the Math team at Ulster Community College, Wolfram MathWorld, and Math is Fun.
- Wikipedia articles on Exponentiation and the Exponential function, describing exponents and their use.
- Video introductions to exponents from Khan Academy and Math Concepts.
- Video explanations of the laws of exponents from Brian McLogan, Math Antics, The Organic Chemistry Tutor, and Mario’s Math Tutoring.

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