Making Sense of Weird Exponents

Exponents are a quick way to represent repeated multiplication. Raising a base number to the power of an exponent means multiplying the base by itself the number of times indicated by the exponent. For example:

102 = 10 × 10 = 100
25 = 2 × 2 × 2 × 2 × 2 = 32
9991 = 999

This definition makes sense when the exponent is a positive integer. But what happens with an exponent of 0, or a negative number, or a fraction?

Becoming one with an exponent of 0

Any value (other than zero) raised to the power of 0 equals 1. For example:

20 = 1
100 = 1
1,4230 = 1

To understand why this rule works, consider the following values of 2 raised to the power of the first few positive integers:

21 22 23 24 25 26
2 4 8 16 32 64

Reading the second row of the table from left to right, every number is twice the previous number. You can continue this pattern indefinitely. Similarly, reading the second row of the table from right to left, every number is half of the next number. So you can continue this pattern as follows:

20 21 22 23 24 25 26
1 2 4 8 16 32 64

This type of pattern holds not just for a base of 2, but for all bases. For example, here is a base of 10:

100 101 102 103 104 105 106
1 10 100 1,000 10,000 100,000 1,000,000

For this reason, every number (except 0) raised to the power of 0 equals 1. To state this rule more formally:

x0 = 1 (when x ≠ 0)

Flipping for negative exponents

To understand exponents of negative integers, continue the table for a base of 2 for a few more columns to the left:

2-4 2-3 2-2 2-1 20 21 22 23 24 25 26
1/16 1/8 1/4 1/2 1 2 4 8 16 32 64

As you can see, the pattern still holds — every number in the bottom row is half the number to its left and twice the number to its right. Notice that every negative exponent of a number is the reciprocal of the corresponding positive exponent. For example:

21 = 2
image0.jpg
22 = 4
image1.jpg
23 = 8
image2.jpg

For this reason, every number raised to a negative integer equals the reciprocal of that number raised to the positive (absolute) value of that integer. To state this rule more formally:

image3.jpg
(when x ≠ 0)

Rooting around for fractional exponents

The rules discussed above outline how to interpret any integer exponent. When an exponent is a fraction, a different approach is needed.

To begin, remember that to multiply two exponential values with the same base, the rule is to add the exponents. For example:

23 × 24 = 27 = 128

Here’s the rule more generally stated:

(xa)(xb) = xa+b

This rule also applies to fractions, so:

image4.jpg

So,

image5.jpg

is a value that, when multiplied by itself, equals 2. That is:

image6.jpg
Because
image7.jpg

This rule works for every positive base, so here’s this rule more generally stated:

image8.jpg
(when x ≥ 0)

This same reasoning works for the definition of other fractions with 1 in the numerator. For example:

image9.jpg

So,

image10.jpg

is a value that, when multiplied by itself 3 times, equals 2. That is:

image11.jpg
Because
image12.jpg

This rule also works for every base, so here it is more generally stated:

image13.jpg
(when x ≥ 0)

Finally, you can extend this reasoning to all fractions. For example:

image14.jpg

You can state this rule for all rational numbers as follows:

image15.jpg
(when n ≠ 0 and x ≥ 0)
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