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:
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.
The properties of exponents can be derived from the definition of exponent.
As an example, let m=2 and n=3.
Since there are five b's multiplied together,
Mathematicians use a negative exponent to mean division, or to mean the reciprocal of a number.
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
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
Here, it is important to note that
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,
But, if n = 2, then
Because the second exponent n=2 mean multiply b3 by itself twice. So,
Exponentiation distributes across multiplication.
Exponentiation distributes across division.
The numerator of a fractional exponent is a power. The denominator is a root.
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