Pronunciation: /ˈɛks.poʊ.nənt/ Explain

An exponent is used to indicate repeated multiplication, which is also called raising a base to a power. For example, 22 means base 2 raised to the power of 2 or 2 multiplied by itself 2 times: 2·2 = 4. 34 means 3·3·3·3 = 81. The process of raising a base to an exponent is called exponentiation.

An exponent can also be called a power. In British english, an exponent is called an index (plural indices).

A value with an exponent that is a unit fraction is called a root or a radical. A special notation is used for roots. The base of the expression is placed inside of a radical sign: Cube root of a, shown with a radical that has 3 in the crook, and a inside the radical.

A negative exponent is used to indicate multiplication by a reciprocal (or multiplicative inverse), which is equivalent to division. So 2-3 = 1/(23) = 1/8.

Properties of Exponents

The properties of exponents can be derived from the definition of exponent.

bm · bn = bm + n As an example, let m = 2 and n = 3.
bm = b2 = b · b,
bn = b3 = b · b · b.
bm · bn = b2 · b3
Since there are five b's multiplied together,
b2 · b3 = (b · b) · (b · b · b) = b5

b<sup>-n</sup> = 1/(b<sup>n</sup>)

Mathematicians use a negative exponent to mean division, or to mean the reciprocal of a number.

b<sup>m</sup>/b<sup>n</sup> = b<sup>m-n</sup>

This says that when we use a negative exponent, we mean the multiplicative inverse, or reciprocal. To see how that works, look at the expression

Because exponents mean repeated multiplication, we can write this expression as
And, because of the commutative property of multiplication we can write this as
But, since anything divided by itself is 1, this becomes
So we can write this as
b<sup>3</sup>·b<sup>-2</sup> = b<sup>3-2</sup> = b<sup>1</sup> = b


To see why this is true, we will start with the right-hand side of the identity, 1. Start with the fact than any number divided by itself is 1, except for 0. So,

But, by the property of dividing by exponents,
We also know that any number less itself is zero, so

(b<sup>n</sup>)<sup>m</sup> = b<sup>m·n</sup>

Here, it is important to note that

(b^m)<n does not equal b^m^n
In the first, we raise b to the m power then raise that result to the n power. In the second, we raise m to the n power and take that result and raise b to that power. These have two have different meanings.

This concept is an extension of the property that bm · bn = bm + n. However, we are dealing with repeated multiplication in both steps. Let's start with bm. Let m = 3. Then,

b3 = b · b · b
But, if n = 2, then
(b3)2 = b3 · b3
Because the second exponent n = 2 mean multiply b3 by itself twice. So,
(b3)2 = b3 · b3 = (b · b · b) · (b · b · b)
(b^m)<n does not equal b^m^n

(ab)n = an · bn

Exponentiation distributes across multiplication.


Exponentiation distributes across division.

a^(m/n)=(nth root of a) to the mth power=nth root of (a to the mth power)

The numerator of a fractional exponent is a power. The denominator is a root.


  1. McAdams, David E.. All Math Words Dictionary, exponent. 2nd Classroom edition 20150108-4799968. pg 76. Life is a Story Problem LLC. January 8, 2015. Buy the book

Cite this article as:

McAdams, David E. Exponent. 4/20/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/e/exponent.html.

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Revision History

4/20/2019: Updated expressions and equations to match new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
7/5/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
2/2/2010: Added "References". (McAdams, David E.)
2/10/2009: Added fractional exponent. (McAdams, David E.)
1/9/2009: Added distributive properties of exponentiation. (McAdams, David E.)
1/4/2009: Added 'More Information'. (McAdams, David E.)
11/25/2008: Changed equations to images. (McAdams, David E.)
2/4/2008: Correct typographical errors. (McAdams, David E.)
8/7/2007: Initial Version. (McAdams, David E.)

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