Introduction

Once you've mastered adding (and subtracting), you're then taught multiplication, which is really just a fast way to add the same number over and over again. Adding can teach you to answer 9 + 9 + 9 + 9, but multiplication turns that problem into 9 * 4, which is easier to write and, once you've mastered multiplication, quicker to solve.

What multiplication is to adding, exponents are to multiplication. With multiplication, you could write out 9 * 9 * 9 * 9, but exponents give you a shorthand notation for that process, writing it as 9⁴. When you write out an exponential equation in a manner like x^y, x is referred to as the base, and y is referred to as the exponent.

Is it really possible to work with exponents in your head? Solomon W. Golomb (This and all other external links in this tutorial will open in a new window) often astounds people by performing seemingly complex exponential problems in his head. In one famous incident, his college biology teacher explained that humans had 24 chromosomes (as was believed at the time), so the number of possibilities was 2²⁴, and jokingly added that everyone knew what number that was. Young Mr. Golomb (now Dr. Golomb) immediately replied that it was 16,777,216. The teacher thought he was trying to be funny, performed the calculation on a calculator, only to find out that 2²⁴ was indeed 16,777,216!

How?

The answer is part pattern recognition, part memorization, and part math. To start, we need to learn the answers to all the exponential expressions X^Y, where both X and Y range from 0 to 10. You might think of these as your exponential tables, as opposed to your multiplication tables.

The thought of memorizing the answers to 121 exponential expressions (11 numbers from 0 to 10 taken to any of the 11 powers from 0 to 10) may sound tough, but before we even begin memorizing answers, we'll start with a few easy-to-remember patterns to cut down on that number.

Base of 0

Taking 0 to any power from 1 to 10 is simple, the answer is 0! 0⁰, however, is a special case. Depending on your source, it's either 0, 1, or undefined. If you're having someone verify this with a calculator, just respond with ERROR.

	Exponential Equation
	Answer to Equation

Exponential Equation	Answer to Equation

00	Indeterminate (Why?)
01	0
02	0
03	0
04	0
05	0
06	0
07	0
08	0
09	0
010	0

Base of 1

1 is even easier to deal with than 0! 1 to any power, even 0, is 1.

	Exponential Equation
	Answer to Equation

Exponential Equation	Answer to Equation

10	1
11	1
12	1
13	1
14	1
15	1
16	1
17	1
18	1
19	1
110	1

Base of 10

Taking 10 to any power from 0 to 10 is easy, as we all learned in school. Write down a 1, look at the exponent, and write down that many zeroes after it! 10²? That's a 1 followed by 2 zeroes, or 100. 10⁹? 1,000,000,000 (1 followed by 9 zeroes). You'll probably want to remember that 10³ is a thousand, 10⁶ is a million, and 10⁹ is one billion.

	Exponential Equation
	Answer to Equation

Exponential Equation	Answer to Equation

100	1
101	10
102	100
103	1,000
104	10,000
105	100,000
106	1,000,000
107	10,000,000
108	100,000,000
109	1,000,000,000
1010	10,000,000,000

Exponent of 0

For any number from 1 to 10 to the power of 0, the answer is always 1 (Why?). As discussed above, 0⁰ is an unusual case.

	Exponential Equation
	Answer to Equation

Exponential Equation	Answer to Equation

00	Indeterminate
10	1
20	1
30	1
40	1
50	1
60	1
70	1
80	1
90	1
100	1

Exponent of 1

First powers are also very easy. Any number to the first power is itself. 5¹? 5! 9¹? 9! 4,326,528¹? 4,326,528!

	Exponential Equation
	Answer to Equation

Exponential Equation	Answer to Equation

01	0
11	1
21	2
31	3
41	4
51	5
61	6
71	7
81	8
91	9
101	10

The Remaining Exponents

With just those few simple patterns above, you've already learned 49 different answers! There's still 72 to go, but that number will quickly be minimized.

Once you're comfortable with these patterns, click here to continue the lesson.