Exponential Expressions, Part 1

Published on Tuesday, October 05, 2010 in , , ,


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 xy, 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!


The answer is part pattern recognition, part memorization, and part math. To start, we need to learn the answers to all the exponential expressions XY, 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
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
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
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
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
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.

Squares and Cubes

Next, you'll need to learn the numbers to the 2nd and 3rd powers. These should be learned so that you know them cold. Not only will knowing these by heart help when giving the answers, but they will also help you handle larger equations later on.

Exponent of 2

Taking a number to the 2nd power is also known as squaring it. If you're still comfortable with your times tables up to 10 times 10, and can recall that 82 is 64 and that 92 is 81, you don't need to memorize the squares. For those who do need to memorize the squares, I've given them to you in the chart below. Since you've learned the answers for 02, 12, and 102 on the previous page, I won't include those in the chart below.

Exponential Equation Answer to Equation
22 4
32 9
42 16
52 25
62 36
72 49
82 64
92 81

Exponent of 3

Taking a number to the 3rd power is also referred to as cubing a number. It's less common to know the cubes offhand than it is for the squares.

Those of you who've put the time in to learn the root extraction feat will have an advantage, as they will already know the cubes by heart! For those who haven't memorized them already, here they are:

Exponential Equation Answer to Equation
23 8
33 27
43 64
53 125
63 216
73 343
83 512
93 729

Learning the Squares and Cubes

To help make sure you remember the squares and cubes, go to the exponent quiz page, and click on the Squares and Cubes button to practice them. Once you have these memorized, you've only got 56 more to go. This means you've already memorized more than half of the 121 answers!

Learning the Remaining Exponents

Before we continue on, you will need to practice with the Link System and the Major System, in order to commit these answers to memory.

Once you're comfortable with the patterns and the memory systems, click here to continue the lesson.

4th and 5th Powers

As mentioned in the previous section, you'll need to be comfortable using the following memory techniques:


Link System
Major System

To convert these exponential exponential expressions for use with the Major System, each equation will be converted into a 2-digit number, with the base in the 10s place, and the exponent as the 1s place. For example, to remember the problem for 64, think of it as 64, and use the mnemonic you developed for 64 (I use JaR). The answer to 64 is 1,296, which translates into hiD hoNey PouCH. Use the link system to link JaR to the phrase hiD hoNey PouCH, and the Major System to convert those words and phrase into numbers. Properly done, given a problem like 64, your thought process should go something like this: 64 = 64 = JaR = hiD hoNey PouCH = 1,296

Does this mean that, for a problem like 610, you'll need to develop a mnemonic for 610? No. To make things easier, all bases to the 10th power will represented by a 0 in the 1s place. For 610, you'll use your mnemmonic for 60, for 710, you'll use your mnemmonic for 70, and so on. Other than this exception, 10th powers are handled like all the other numbers. If you've practiced my 400 digits of Pi feat, this will be familiar to you.

In all the charts, I'll be including the exponential equation, the key number to use for the mnemonic, the equation answer, the key number mnemonic, and the answer mnemonic. If you've developed mnemonic for the numbers from 1-100 different than the ones I employ in this tutorial, feel free to use those. The important thing is to make sure that you're able to link them to the answer mnemonic, so that you get the correct answer.

Occasionally, you'll see words in the mnemonics highlighted. These are ones that I believe not everybody will be familiar with, and lead to Wikipedia links explaining exactly what they mean. All these links will open up in a new window.

As the numbers get larger and the charts have more columns, you may find that it's more difficult to read the mnemonics. If so, you can simply re-adjust the widths of the columns in the tables below by clicking on the dividing line and dragging it left or right as desired.

Exponent of 4

Unlike taking a number to the 2nd or 3rd power, taking a number to a 4th or higher power don't have any commonly accepted name like squaring or cubing. They're simply referred to as taking the number to the given power (e.g., 4th power, 5th power).

Equation Key Number Answer to Equation Key Mnemonic Answer Mnemonic
24 24 16 NeRo DaSH
34 34 81 MoweR FeeT
44 44 256 RoweR iNN LoDGe
54 54 625 LuRe CHaNNeL
64 64 1,296 JaR hiD hoNey PouCH
74 74 2,401 CaR New aRReST
84 84 4,096 FiRe hiRe SPeeCH
94 94 6,561 BeeR hooCH LeaSHeD

Exponent of 5

Here are the numbers to the 5th power:

Equation Key Number Answer to Equation Key Mnemonic Answer Mnemonic
25 25 32 NaiL MoNey
35 35 243 MuLe NoRM
45 45 1,024 RoLL Die SceNaRio
55 55 3,125 LiLy Mow TuNNeL
65 65 7,776 JaiL Go Key CaGe
75 75 16,807 CoaL DiTCH PHySiQue
85 85 32,768 FiLe MeanN KeY SHaVe
95 95 59,049 BeLL LaB SyRuP

Learning Up to the 5th Power

Let's take a break here. Make sure you have all the numbers up to the 5th power memorized by going to the exponent quiz page, and click on the 4th and 5th Powers button to practice them.

Learning the Next Powers

Once you're comfortable taking numbers up to their 5th powers, and you've taken a break, click here to continue the lesson. You've already learned 81 exponents, so there are only 40 to go!

6th Through 8th Powers

Since you've got the basic idea by now, we'll move right to the next three sets of powers.

Exponent of 6

Here are the numbers to the 6th power:

Equation Key Number Answer to Equation Key Mnemonic Answer Mnemonic
26 26 64 NoTCH CHeRRy
36 36 729 MaTCH CaNoPy
46 46 4,096 RoaCH hiRe SPeeCH
56 56 15,625 LeeCH DiaL CHaNNeL
66 66 46,656 JuDGe RiDGe GeoLoGy
76 76 117,649 CaGe iDioTiC CHiRP
86 86 262,144 FiSH NoTioN DRieR
96 96 531,441 BeaCH aLMighTy RewaRD

Exponent of 7

Here are the numbers to the 7th power:

Equation Key Number Answer to Equation Key Mnemonic Answer Mnemonic
27 27 128 NeCK haD kNiFe
37 37 2,187 MuG New DoVeKie
47 47 16,384 RoCK DiTCH MoVeR
57 57 78,125 LoG GooFy TuNNeL
67 67 279,936 JaCK youNG Boy Buy MuCH
77 77 823,543 CooK VeNoM aLaRM
87 87 2,097,152 FoG New SPooKy heaDLiNe
97 97 4,782,969 BaG aRe GiVeN By SHoP

Exponent of 8

Here are the numbers to the 8th power:

Equation Key Number Answer to Equation Key Mnemonic Answer Mnemonic
28 28 256 kNiFe iNN LoDGe
38 38 6,561 MoVie hooCH LeaSHeD
48 48 65,536 RooF SHaLe oiL MaTCH
58 58 390,625 LaVa iMPaSSe CHaNNeL
68 68 1,679,616 CHeF eaT, CHeCKuP, waSH DiSH
78 78 5,764,801 CaVe hoLe CaSHieR ViSiT
88 88 16,777,216 FiFe TeaCH, Cue GiG, NighT SHow
98 98 43,046,721 BeeF aRoMa, hiSS, RiCH weeKeND

Learning Up to the 8th Power

It's time for another break. To make sure you have all the numbers up to the 8th power memorized, and to reinforce all the ones you've learned previously, go to the exponent quiz page, and click on the 6th, 7th, and 8th Powers button to practice them.

Learning the 9th and 10th Powers

You're almost there! If you can remember all the exponents so far, click here to learn the last two powers!

9th Through 10th Powers

Here are the final two powers. Get these, and you'll know all the exponents from 00 up to 1010!

Exponent of 9

Here are the numbers to the 9th power:

Equation Key Number Answer to Equation Key Mnemonic Answer Mnemonic
29 29 512 kNoB LowDowN
39 39 19,683 MoP DeeP huGe FoaM
49 49 262,144 RoPe NoTioN DRieR
59 59 1,953,125 LiP weT BaLM TuNNeL
69 69 10,077,696 SHiP ToSS SeaQuaKe hoDGePoDGe
79 79 40,353,607 CuP RiCe MeaL, haM, CHeeSe, eGG
89 89 134,217,728 FoB aDMiRe aNTiQue CoNVoy
99 99 387,420,489 BiB MoVe eGG - RuNS heRe oFF Boy

Exponent of 10

Finally, here are the numbers to the 10th power:

Equation Key Number Answer to Equation Key Mnemonic Answer Mnemonic
210 20 1,024 NoSe Die SceNaRio
310 30 59,049 MouSe LaB SyRuP
410 40 1,048,576 RoSe ToW - SeRVe LuGGaGe
510 50 9,765,625 LiCe huB eGGSHeLL CHaNNeL
610 60 60,466,176 CHeeSe CHeeSe ouR JuDGe woulD GauGe
710 70 282,475,249 CaSe New PHoNy oRaCLe NeaRBy
810 80 1,073,741,824 FuSe weD, aSK My GReaT wiFe NeaR
910 90 3,486,784,401 BuS My ReFuGe, Go FoR youR SeaT

Learning Up to the 10th Power

Memorize these, and practice them by going to the exponent quiz page, and click on the 9th and 10th Powers button.

Going beyond 1010

You could certainly stop here, and still have an impressive new ability. However, you can go on to learn to perform even more challenging exponential expressions, and without memorizing any more answers!

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