Tackling Exponents and Square Roots
Chapter 11: Tackling Exponents and Square Roots
“Cubic feet” is a common measure of commercial and consumer refrigerator capacity. When welders talk about 40 CF, 60 CF, and 80 CF tanks, they aren’t referring to the size of the tank but rather to the amount of compressed weld- ing gas the tank holds.
In the world of science, you may see metric volume measurements such as cubic centimeters (cm3); however, in the lab or hospital, a cubic centimeter may be abbreviated as cc or referred as a milliliter (ml).
Moving beyond 2 or 3
Powers don’t stop with just the second and third power. The most ambitious powers want to be bigger than 2 or 3 because higher powers (in addition to being generally revered) are a very compact way of expressing very large and very small numbers, which we discuss further in the “Powers with base 10”
and “Powers with base 2” sections later in this chapter.
Table 11-1 shows you an example of a simple expression representing a large number:
Table 11-1 Conveying Large Numbers with Exponents
Exponent Math Result
22 2 × 2 4
23 2 × 2 × 2 8
24 2 × 2 × 2 × 2 16
25 2 × 2 × 2 × 2 × 2 32
26 2 × 2 × 2 × 2 × 2 × 2 64
27 2 × 2 × 2 × 2 × 2 × 2 × 2 128
228 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
268,435,456
In Table 11-1, the base (2) doesn’t change, but the exponent does. Notice how quickly a small three-character expression (228) represents a very large number.
Such large numbers are easier to manipulate when you express them in such a compact form. The same is true with SI units (the International System of Units, which you commonly call the metric system). A little bit of writing gets you a whole lot of number. The range of metric terms (for liters, for example) goes from the very tiny yoctoliter (10–24, or 1/1,000,000,000,000,000,000,000,000) to the very large yottaliter (1024, or 1,000,000,000,000,000,000,000,000).
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Different faces of special bases
When you work with exponentiation, you can use almost any base and almost any power. In algebra, you see terms whose values aren’t known — for exam- ple, ab — and they can represent anything until you know the solution.
However, you need to know about a few special bases and powers. A couple of them drive the world these days, and others are just oddities (unless you’re a theoretical physicist). Here are tonight’s contestants on Dancing with the Power and Base Stars; we discuss them in the following sections:
✓ Powers with base 10: Much like basketball and ice hockey power for- wards, base 10 is the star of the show.
✓ Powers with base 2: This base has been rising fast since the dawn of the computer age.
✓ Powers with base 1: This dull base only has one trick.
✓ Powers with base 0: This base is also dull, but with a hint of controversy.
✓ Powers with base –1: A base with only two results.
✓ Powers of 1 and 0: These two don’t deal with bases but with rather unusual properties of certain powers.
Powers with base 10
You grow up counting to 10. Later, you count to 100, and one day on a dull afternoon, you count to 1,000. These three numbers are all multiples of (and therefore powers of) 10. When you express them with powers, 10 becomes 101, 100 becomes 102, and 1,000 becomes 103.
In middle school and high school, you go a step further. Put a 1 in front of a 10 raised to a power and you have scientific notation. The numbers 10, 100, and 1,000 become 1 × 101, 1 × 102, and 1 × 103.
Powers of 10 are prominent in the lab and in the observatory, whether you measure the number of molecules in a reagent, the diameter of an atom’s nucleus or the distance to a star. For example, NASA says the distance to Proxima Centauri is about 39,900,000,000,000 kilometers. It’s a lot easier to write this distance as 39.9 × 1012 kilometers.
Using powers of 10 is a much easier way to do math on very large or very small numbers. In exponentiation, the equivalent of a decimal shift is increas- ing or decreasing the power. A one-decimal shift to the right gets you a tenfold
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Chapter 11: Tackling Exponents and Square Roots
increase in the value of a number. That is, to multiply a number with a base of 10 by 10, just increase its exponent by 1. A one-decimal shift to the left produces a tenfold decrease in the value of a number. To divide a number with a base of 10 by 10, decrease its exponent by 1. These shifts mean you’re changing the order of magnitude.
Look at this example:
10 × 1,000 = 10,000 or 10 × 103 = 104 That’s increasing the number by a factor of 10.
You can also use negative powers of 10 — they represent division. A milliliter is of a liter, and you express that as 10–3 liters.
Powers with base 2
Powers with base 2 are at the very core of computing. From the earliest elec- tronic computers to the latest ones you need for your work, the internal math is base 2 math.
Counting to ten was a pretty easy task for our ancestors, and civilizations have risen and fallen while they used decimal math. Trouble is, counting to ten is very complicated for a computer. Counting to 2, however, isn’t bad. A microprocessor can multiply by 2 or divide by 2 just by doing a bit shift. Very efficient. If you’re a microprocessor, you know two states (1 for “on” and 0 for
“off”), and that’s all you need.
You as a professional use the powers of base 2 in two ways:
✓ If you work with computers professionally (in information technology or as a computer tech), various base 2 numbers are your daily compan- ions. Those numbers include
• Processor speed in Hz
• RAM in gigabytes
• Disk capacity in megabytes, gigabytes, and terabytes
• Network speed in megabits and gigabits
• Video board memory in megabytes
✓ If you buy or use computers as part of your work, you’ll see base 2 num- bers (or their abbreviations) all the time.
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Table 11-2 lists some common abbreviations and their numeric representa- tions in base 2 and base 10 (covered in the preceding section):
Table 11-2 Common Base 2 Computer Abbreviations
Term Base 2 Base 10
Megabyte 220 1,048,576
Gigabyte 230 1,073,741,824
Terabyte 240 1,099,511,627,776
IPv6 address space
2128 340,282,366,920,938,463,463,374,607,431,768,211,456
Note that the name megabyte actually means 1 million bytes. However, the real number of bytes in a megabyte is 1,048,576, and everybody goes along with the convention.
Did you know that the world may be running out of Internet addresses? The current IPv4 system has only 4,294,967,296 (232) addresses available. The pro- posed IPv6 system would have 2128 addresses. That’s a big number.
Powers with base 1
This topic is special. It won’t come up in your work, but you need to know about it to get the whole picture. What happens when you elevate 1 to vari- ous powers? Use this handy table to see the answer:
Spoken Form Written Form Math Result
1 to the first 11 1 × 1 1
1 squared 12 1 × 1 1
1 cubed 13 1 × 1 × 1 1
1 to the eighth power 18 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1 1 Yes, 1 to any power equals 1.
Powers with base 0
Powers with base 0 have produced a controversy, and you won’t encounter them in your work. However, you need to know about them, so here are the rules, straight from multiple online sources.
✓ If the exponent is positive, the power of zero is zero:
0n = 0, no matter how large the exponent.
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Chapter 11: Tackling Exponents and Square Roots
✓ If the exponent is negative, the power of zero (0n, where n < 0) is called
“undefined,” because division by zero is implied and that’s impossible.
0–n = undefined.
✓ If the exponent is zero, some mathematicians define it as one and others leave it undefined.
00 = 1 or may be undefined. Either choice is a safe bet for you.
Powers with base (–1)
A base of (–1) isn’t much to shout about, but it’s twice as exciting as powers with a base of 1 — it has two rules.
✓ When the exponent n even, −1n = 1 ✓ When the exponent n is odd, −1n = −1
Powers of 1 and 0
Exponentiation has a couple of special conditions dealing with the powers (not the bases) of 1 and 0. This section introduces you to them, but the fol- lowing section shows you them in action.
The math of exponentiation shows that any number raised to the power of 1 is itself, as the following examples show:
11 = 1 101 = 10 7561 = 756
2,568,145,2591 = 2,568,145,259
Any number raised to the power of 0 is 1, as indicated in the following examples:
10 = 1 100 = 1 7560 = 1
2,568,145,2590 = 1
Exponentiation math
Exponentiation math is fast, fair, and friendly. To multiply two numbers with the same base, you add the powers. To divide the numbers, subtract the powers. A term with an exponent represents repeated multiplication.
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When you multiply two terms with exponents, you see that the result is iden- tical to adding the exponents.
For example, 52 is 5 × 5 and 53 is 5 × 5 × 5.
That is, 52 is 25 and 53 is 125. If you multiply 52 by 53, you’re multiplying 25 by 125. The answer is 3,125.
What if you add the exponents 2 and 3? The result is 55 or 5 × 5 × 5 × 5 × 5.
And the result of doing the multiplication is 3,125. The answer is the same:
3,125.
The same idea is true when you divide; you just subtract the powers.
Use the figures from the multiplication example and divide 53 by 52.
What if you just subtract the exponents? You get the same result.
Now if the result turns out to be 50, the answer is 1, because any base raised to the power of 0 is 1.
If the result turns out to be a negative exponent (such as 5–1), it’s a reciprocal.
The general rule is
Using 5 as an example:
The answer is
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Chapter 11: Tackling Exponents and Square Roots
And to see this process in a longer form, divide 52 by 53.
Again, the answer is
Getting Back to Your(Square)Roots
You know how to square numbers. The subject may have come up in elemen- tary school very soon after you learned multiplication, and then again in high school math and maybe in community college math. And multiplying to get a square value comes up in several chapters in this book, including this very one. The earlier section “The basics of the base” shows you that a square is another way of describing a number raised to the power of 2. For example 5 squared is also 52 and equals 25.
But what if you have the square and need to find out what it’s the square of?
The inverse of squaring a number is finding a number’s square root. A square root is all about finding the value of the base when you know only the result of squaring the value. A square root operation has a symbol. The following symbol shows a problem where you’re looking for the square root of 16:
So what’s the square root of 16? It’s 4. When you were younger, you prob- ably studied the easy ones: 4 (answer, 2), 9 (answer, 3), 16 (answer, 4) and 25 (answer, 5). That’s great, but what happens when you need to find the square root of 625? You need a math solution, not memory. Luckily, the following sections give you three methods for finding square roots: the hard way, the easy way, and the effortless way.
Note: Truth in advertising: You may not find a lot of opportunities to use square root math in your trade. For example, if a cement mason knows the area of a circular patio, he or she can use a square root formula to find the patio’s radius and diameter. But that’s a bit backward because the mason usually starts with the linear dimensions (such as the radius) and then comes up with the area to calculate the volume of the concrete pour.
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Why are they called roots, anyway? Maybe because, like a plant’s roots, they’re basic and lie below the surface. Two sources say that in math, the root of a number x is any number that, when repeatedly multiplied by itself, even- tually yields x. A square root is a root where the number is multiplied by itself once. A cube root is where the number is multiplied by itself twice. Higher- order roots are possible, too.
Square roots the hard way
A manual method for finding square roots does exist, but it’s not for the faint- hearted. High schools taught it in the 1960s, and even then only in the accel- erated math classes. It’s cumbersome and slow, although it’s accurate. You can find the method on the Internet, but the chances are good that you won’t find it to be a good use of your time.
Square roots the easy way
Use the technique of successive approximation (also called guessing) with the help of a calculator. This method is the thinking person’s guessing.
For example, if you want to find the square root of 19 to three decimal places, take the following approach:
1. Examine the situation.
You know that the square root of 25 is 5. The result 25 is too high, so the root 5 is too high. You know that the square root of 16 is 4. That result (16) is too low. The answer you want is somewhere between 4 and 5.
2. Split the difference between your too-high and too-low numbers.
Try punching 4.5 into your calculator and squaring it. The answer is 20.25. That’s a little high.
3. If your result isn’t quite right, try again, splitting the difference between your most recent guess and a lower number (if your guess was too high) or a higher number (if your guess was too low).
Split the difference between 4 and 4.5. Enter 4.25 on your calculator and square it. The answer is 18.0625, so you’re getting closer.
4. Repeat Step 3 until you get the most accurate answer you can.
Split the difference between 4.25 and 4.5. With 4.375, you get 19.140625, which is still just a little high. Try 4.3, 4.335, 4.355, 4.357, and 4.358. The last answer is 18.992. Close enough, though you can keep approximating if you like.
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Chapter 11: Tackling Exponents and Square Roots
Square roots the effortless way
The effortless way to calculate a square root doesn’t help your math skills, but it does get the job done. Pick one of two methods:
✓ Use your pocket calculator, scientific calculator, or smartphone. It may have a square root key:
✓ Use Microsoft Excel or Open Office Calc. The Excel square root function is =SQRT(nnn), where nnn is the number whose square root you want to find.
Example: Finding the Bytes On a Disk
You’re installing disks with an advertised capacity of 320 gigabytes (GB). Your curiosity gets the better of you and you ask, “How many bytes is that?” You can solve this problem in two ways, and both are simple:
✓ Method 1: Take the known decimal number for 1 GB and multiply it by the number of bytes advertised. Table 11-2 earlier in this chapter tells you that known number is 1,073,741,824 bytes/gigabyte. One gigabyte is also 230 bytes.
The answer is 343,597,383,680 bytes (although don’t get too excited — the disk will have less capacity after it’s formatted).
✓ Method 2: Take the known base 2 number for a gigabyte (also listed in Table 11-2) and multiply it by the number of gigabytes.
This answer is technically correct, but it probably doesn’t satisfy your urge to see the answer in big base 10 digits. If you expand the factor of 230 to become 1,073,741,824 bytes, you’re repeating method 1.
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