Exponent

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.

PropertyExplanation

b<sup>m</sup>·b<sup>n</sup> = b<sup>m+n</sup>

As an example, let m=2 and n=3.
Then

b<sup>m</sup> = b<sup>2</sup> = b·b,
and
b<sup>n</sup> = b<sup>3</sup> = b·b·b.
So
b<sup>m</sup>·b<sup>n</sup> = b<sup>2</sup>·b<sup>3</sup>
Since there are five b's multiplied together,
b·b · b·b·b = b<sup>5</sup>

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

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

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

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,

1=(b^m)/(b^m)
But, by the property of dividing by exponents,
(b^m)/(b^m)=b^(m-m)
We also know that any number less itself is zero, so
b^(m-m)=b^0
So,
(b<sup>m</sup>)<sup>n</sup>=b<sup>m·n</sup>

(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. The 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,

b^3=b*b*b
But, if n = 2, then
(b^3)^2=(b^3)*(b^3)
Because the second exponent n=2 mean multiply b3 by itself twice. So,
b^3)^2=(b^3)*(b^3)=b^(3+3)=b^6=b^(2*3)
So,
(b<sup>m</sup>)<sup>n</sup>=b<sup>m·n</sup>

(a*b)^n=a^n*b^n

Exponentiation distributes across multiplication.

(a/b)^n=(a^n)/(b^n)

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.

Cite this article as:

McAdams, David E. Exponent. 7/10/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/e/exponent.html.

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Revision History

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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