Exponents

Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 5³. The "exponent", being 3 in this example, stands for however many times the value is being multiplied. The thing that's being multiplied, being 5 in this example, is called the "base".
This process of using exponents is called "raising to a power", where the exponent is the "power". The expression "5³" is pronounced as "five, raised to the third power" or "five to the third". There are two specially-named powers: "to the second power" is generally pronounced as "squared", and "to the third power" is generally pronounced as "cubed". So "5³" is commonly pronounced as "five cubed".
When we deal with numbers, we usually just simplify; we'd rather deal with "27" than with "3³". But with variables, we need the exponents, because we'd rather deal with "x⁶" than with "xxxxxx".
Exponents have a few rules that we can use for simplifying expressions.

• Simplify (x³)(x⁴)   Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

To simplify this, I can think in terms of what those exponents mean. "To the third" means "multiplying three copies" and "to the fourth" means "multiplying four copies". Using this fact, I can "expand" the two factors, and then work backwards to the simplified form:
(x³)(x⁴) = (xxx)(xxxx)           = xxxxxxx           = x⁷

Note that x⁷ also equals x⁽³⁺⁴⁾. This demonstrates the first basic exponent rule: Whenever you multiply two terms with the same base, you can add the exponents:

( x ^m ) ( x ⁿ ) = x^{( m + n )}

However, we can NOT simplify (x⁴)(y³), because the bases are different: (x⁴)(y³) = xxxxyyy = (x⁴)(y³). Nothing combines.

• Simplify (x²)⁴

Just as with the previous exercise, I can think in terms of what the exponents mean. The "to the fourth" means that I'm multiplying four copies of x²:
(x²)⁴ = (x²)(x²)(x²)(x²)        = (xx)(xx)(xx)(xx)        = xxxxxxxx        = x⁸

Note that x⁸ also equals x^{( 2×4 )}. This demonstrates the second exponent rule: Whenever you have an exponent expression that is raised to a power, you can multiply the exponent and power:

( x^m ) ⁿ = x ^{m n}

If you have a product inside parentheses, and a power on the parentheses, then the power goes on each element inside. For instance, (xy²)³ = (xy²)(xy²)(xy²) = (xxx)(y²y²y²) = (xxx)(yyyyyy) = x³y⁶ = (x)³(y²)³. Another example would be:

Warning: This rule does NOT work if you have a sum or difference within the parentheses. Exponents, unlike mulitiplication, do NOT "distribute" over addition.
For instance, given (3 + 4)², do NOT succumb to the temptation to say "This equals 3² + 4² = 9 + 16 = 25", because this is wrong. Actually, (3 + 4)² = (7)² = 49, not 25. When in doubt, write out the expression according to the definition of the power. Given (x – 2)², don't try to do this in your head. Instead, write it out: "squared" means "times itself", so (x – 2)² = (x – 2)(x – 2) = xx – 2x – 2x + 4 = x² – 4x + 4.
The mistake of erroneously trying to "distribute" the exponent is most often made when the student is trying to do everything in his head, instead of showing his work. Do things neatly, and you won't be as likely to make this mistake.
There is one other rule that may or may not be covered at this stage:

Anything to the power zero is just "1".

This rule is explained on the next page. In practice, though, this rule means that some exercises may be a lot easier than they may at first appear:

• Simplify [(3x⁴y⁷z¹²)⁵ (–5x⁹y³z⁴)²]⁰ Who cares about that stuff inside the square brackets? I don't, because the zero power on the outside means that the value of the entire thing is just 1.
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• A negative exponent just means that the base is on the wrong side of the fraction line, so you need to flip the base to the other side. For instance, "x^–2" (ecks to the minus two) just means "x², but underneath, as in 1/(x²)".
  
  • Write x^–4 using only positive exponents.
    
  • Write x² / x^–3 using only positive exponents.
    
  • Write 2x^–1 using only positive exponents.
     
  Note that the "2" above does not move with the variable; the exponent is only on the "x".
  
  • Write (3x)^–2 using only positive exponents.
    
  Unlike the previous exercise, the parentheses meant that the negative power did indeed apply to the three as well as the variable.
  
  • Write (x^–2 / y^–3)^–2 using only positive exponents.
    
    This one can also be done as:   Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved
    
    
  Since exponents indicate multiplication, and since order doesn't matter in multiplication, there will often be more than one sequence of steps that will lead to a valid simplification of a given exercise. Don't worry if the steps in your homework look quite different from the steps in a classmate's homework. As long as your steps were correct, you should both end up with the same answer.
  

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  By the way, now that you know about negative exponents, you can understand the logic behind the "anything to the power zero" rule:
  
  Anything to the power zero is just "1".
  Why is this so? There are various explanations. One might be stated as "because that's how the rules work out." Another would be to trace through a progression like the following:
  
  3⁵ = 3⁶ ÷ 3 = 3⁶ ÷ 3¹ = 3^6–1 = 3⁵= 243 3⁴ = 3⁵ ÷ 3 = 3⁵ ÷ 3¹ = 3^5–1 = 3⁴= 81 3³ = 3⁴ ÷ 3 = 3⁴ ÷ 3¹ = 3^4–1 = 3³= 27 3² = 3³ ÷ 3 = 3³ ÷ 3¹ = 3^3–1 = 3²= 9 3¹ = 3² ÷ 3 = 3² ÷ 3¹ = 3^2–1 = 3¹= 3
  Then logically 3⁰ = 3¹ ÷ 3¹ = 3^1–1 = 3⁰ = 1.
  A negative-exponents explanation of the "anything to the zero power is just 1" might be as follows:
  
  m⁰ = m^{(n – n)} = mⁿ × m^–n = mⁿ ÷ mⁿ = 1
  ...since anything divided by itself is just "1".
  Another comment: Please don't ask me to "define" 0⁰. There are at least two ways of looking at this quantity:
  
  • Anything to the zero power is "1", so 0⁰ = 1.
  • Zero to any power is zero, so 0⁰ = 0.
  As far as I know, the "math gods" have not yet settled on a "definition" of 0⁰. In fact, in calculus, "0⁰" will be called an "indeterminate form". If this quantity comes up on class, don't assume: ask your instructor what you should do with it.
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By using exponents, we can reformat numbers. For very large or very small numbers, it is sometimes simpler to use "scientific notation" (so called, because scientists often deal with very large and very small numbers). The format for writing a number in scientific notation is fairly simple: (first digit of the number) followed by (the decimal point) and then (all the rest of the digits of the number), times (10 to an appropriate power). The conversion is fairly simple.
• Write 124 in scientific notation. 
This is not a very large number, but it will work nicely for an example. To convert this to scientific notation, I first write "1.24". This is not the same number, but (1.24)(100) = 124 is, and 100 = 10². Then, in scientific notation, 124 is written as 1.24 × 10².
Actually, converting between "regular" notation and scientific notation is even simpler than I just showed, because all you really need to do is count decimal places.
• Write in decimal notation:  3.6 × 10¹²
Since the exponent on 10 is positive, I know they are looking for a LARGE number, so I'll need to move the decimal point to the right, in order to make the number LARGER. Since the exponent on 10 is "12", I'll need to move the decimal point twelve places over. First, I'll move the decimal point twelve places over. I make little loops when I count off the places, to keep track:
Then I fill in the loops with zeroes:   Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved
In other words, the number is 3,600,000,000,000, or 3.6 trillion
Idiomatic note: "Trillion" means a thousand billion — that is, a thousand thousand million — in American parlance; the British-English term for the American "billion" would be "a milliard", so the American "trillion" (above) would be a British "thousand milliard".
• Write 0.000 000 000 043 6 in scientific notation.
In scientific notation, the number part (as opposed to the ten-to-a-power part) will be "4.36". So I will count how many places the decimal point has to move to get from where it is now to where it needs to be:
Then the power on 10 has to be –11: "eleven", because that's how many places the decimal point needs to be moved, and "negative", because I'm dealing with a SMALL number. So, in scientific notation, the number is written as 4.36 × 10^–11
• Convert 4.2 × 10^–7 to decimal notation.
Since the exponent on 10 is negative, I am looking for a small number. Since the exponent is a seven, I will be moving the decimal point seven places. Since I need to move the point to get a small number, I'll be moving it to the left. The answer is 0.000 000 42
• Convert 0.000 000 005 78 to scientific notation.
This is a small number, so the exponent on 10 will be negative. The first "interesting" digit in this number is the 5, so that's where the decimal point will need to go. To get from where it is to right after the 5, the decimal point will need to move nine places to the right. Then the power on 10 will be a negative 9, and the answer is 5.78 × 10^–9
• Convert 93,000,000 to scientific notation.
This is a large number, so the exponent on 10 will be positive. The first "interesting" digit in this number is the leading 9, so that's where the decimal point will need to go. To get from where it is to right after the 9, the decimal point will need to move seven places to the left. Then the power on 10 will be a positive 7, and the answer is 9.3 × 10⁷
Just remember: However many spaces you moved the decimal, that's the power on 10. If you have a small number (smaller than 1, in absolute value), then the power is negative; if it's a large number (bigger than 1, in absolute value), then the exponent is positive. Warning: A negative on an exponent and a negative on a number mean two very different things! For instance:
–0.00036 = –3.6 × 10^–4   0.00036 = 3.6 × 10^–4   36,000 = 3.6 × 10⁴ –36,000 = –3.6 × 10⁴
Don't confuse these!

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You might be asked to multiply and divide numbers in scientific notation. I've never really seen the point of this, since, in "real life", you'd be dealing with these messy numbers by using a calculator, but here's the process.
• Simplify and express in scientific notation: (2.6 × 10⁵) (9.2 × 10^–13)
Since I'm multiplying, I can move things around and simplify some of this stuff easily:
(2.6 × 10⁵) (9.2 × 10^–13)     = (2.6)(10⁵)(9.2)(10^–13)     = (2.6)(9.2)(10⁵)(10^–13)     = (2.6)(9.2)(10^5–13)     = (2.6)(9.2)(10^–8)
Now I have to deal with the 2.6 times 9.2, remembering to convert to scientific notation:
2.6 × 9.2 = 23.92 = 2.392 × 10 = 2.392 × 10¹
Putting it all together, I have:
(2.6 × 10⁵) (9.2 × 10^–13)     = (2.6)(9.2)(10^–8)     = (2.392 × 10¹)(10^–8)     = (2.392)(10¹)(10^–8)     = (2.392)(10^1–8)     = 2.392 × 10^–7
Then (2.6 × 10⁵) (9.2 × 10^–13) = 2.392 × 10^–7
Dividing numbers in scientific notation works about the same way.
• Simplify and express in scientific notation:  (1.247 × 10^–3) ÷ (2.9 × 10^–2)
First, I'll deal with the exponents:
(1.247 × 10^–3) ÷ (2.9 × 10^–2)     = (1.247 ÷ 2.9) (10^–3 ÷ 10^–2)     = (1.247 ÷ 2.9) (10^–3 × 10²)     = (1.247 ÷ 2.9) (10^–1)
Now I'll deal with the division:
1.247 ÷ 2.9 = 0.43 = 4.3 × 10^–1
Putting it all together, I get:
(1.247 × 10^–3) ÷ (2.9 × 10^–2)     = (1.247 ÷ 2.9) (10^–1)     = (4.3 × 10^–1) (10^–1)     = (4.3)(10^–1)(10^–1)     = (4.3)(10^–2)     = 4.3 × 10^–2
So the answer is:  (1.247 × 10^–3) ÷ (2.9 × 10^–2) = 4.3 × 10^–2
If you are required to do problems like these, remember that you can always check your answers in your calculator. For instance, entering "1.247 EE –3 / 2.9 EE –2" on my calculator returns "0.043", which equals 4.3 × 10^–2 in scientific notation. If you have to do a lot of these problems, you may find it useful to set your calculator to display all values in scientific notation. Check your owner's manual for instructions.
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• "Engineering" notation is very similar to scientific notation, except that the power on ten can only be a multiple of three. In this way, numbers are always stated in terms of thousands, millions, billions, etc. For instance, 13,460,972 is thirteen million and some. In the newspaper, it would probably be abbreviated as "13.5 million". In engineering notation, you would move the decimal point six places to the left to get 13.460972 × 10⁶. Once you get used to this notation, you recognize that 10⁶ means "millions", so you would see right away that this is around 13.5 million. Every time a newspaper refers to some number of millions or billions or trillions, rather than writing out the whole number with all the zeroes, it is, in effect, using engineering notation.
  
  • Express 472,690,128,340 in engineering notation.
  This is a twelve-digit number. I need to move the decimal point from the end of the number toward the beginning of the number, but I must move it in steps of three decimal places. In this case, I must move the decimal point to between the 2 and the 6, because this will leave nine digits (and nine is a multiple of 3) after the decimal point, and no more than three digits before the decimal point. Then the answer is:
  472.690128340 × 10⁹, or 472.7 billions.
  • Express 83,201 in engineering notation.
  I need to move the decimal point over to the left in sets of three digits. I can't move the decimal point any further than to the left of the 2, which is three places, so the answer is:
  83.201 × 10³, or 83.201 thousands.
  • Express 0.000 063 8 in engineering notation.
  I need to move the decimal point over in sets of three. If I move the decimal point to the right three places, I'll be left with "0.0638", which won't do. If I move the decimal point to the right nine places, I'll get "63800", which is too many digits. So I need to move the decimal point six places. Since this started out as a small number, the power on 10 will be negative, and the answer is:   Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved
  63.8 × 10^–6, or 63.8 millionths.
  • Express 0.397 53 in engineering notation.
  I need to move the decimal point to the right three places. Since this started as a small number, the power on 10 will be negative:
  397.53 × 10^–3, or 397.53 thousandths.
  You should notice that, in engineering notation, it is perfectly okay to have more than one digit to the left of the decimal point; in fact, you should expect to have something other than always only one digit. Just make sure that the power on 10 is a multiple of three.
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• You already know of one relationship between exponents and radicals: the appropriate radical will "undo" an exponent, and the right power will "undo" a root. For example:
  
  
  But there is another relationship (which, by the way, can make computations like those above much simpler): For the square (or "second") root, we can write it as the one-half power, like this:
  
  
  ...or:   Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved
  
  
  The cube (or "third") root is the one-third power:
  
  
  The fourth root is the one-fourth power:
  
  
  The fifth root is the one-fifth power; and so on.
  Looking at the first examples, we can re-write them like this:
  
  
  You can enter fractional exponents on your calculator for evaluation, but you must remember to use parentheses. If you are trying to evaluate, say, 15^(4/5), you must put parentheses around the "4/5", because otherwise your calculator will think you mean "(15 ⁴) ÷ 5".
  Fractional exponents allow greater flexibility (you'll see this a lot in calculus), are often easier to write than the equivalent radical format, and permit you to do calculations that you couldn't before. For instance:
  
  
  Whenever you see a fractional exponent, remember that the top number is the power, and the lower number is the root (if you're converting back to the radical format). For instance:
  
  
  By the way, some decimal powers can be written as fractional exponents, too. If you are given something like "3^5.5", recall that 5.5 = ¹¹/₂, so:
  
  3^5.5 = 3^11/2
  Generally, though, when you get a decimal power (something other than a fraction or a whole number), you should just leave it as it is, or, if necessary, evaluate it in your calculator. For instance, 3 ^pi , where pi is the number approximately equal to 3.14159, cannot be simplified or rearranged as a radical.
  

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  A technical point: When you are dealing with these exponents with variables, you might have to take account of the fact that you are sometimes taking even roots. Think about it: Suppose you start with the number –2. Then:
  
  
  In other words, you put in a negative number, and got out a positive number! This is the official definition of absolute value:
  
  
  (Yeah, I know: they never told you this, but they expect you to know somehow, so I'm telling you now.) So if they give you, say, x^3/6, then x had better not be negative, because x³ would still be negative, and you would be trying to take the sixth root of a negative number. If they give you x^4/6, then a negative x becomes positive (because of the fourth power) and is then sixth-rooted, so it becomes | x |^2/3 (by reducing the fractional power). On the other hand, if they give you something like x^4/5, then you don't have to care whether x is positive or negative, because a fifth root doesn't have any problem with negatives. (By the way, these considerations are irrelevant if your book specifies that you should "assume all variables are non-negative".)
  

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• A technology point: Calculators and other software do not compute things the way people do; they use pre-programmed algorithms. Sometimes the particular method the calculator uses can create difficulties in the context of fractional exponents.
  For instance, you know that the cube root of –8 is –2, and the square of –2 is 4, so (–8)(2/3) = 4. But some calculators return a complex value or an error message, as is the case with one of my graphing calculators:
  If you enter "=(–8)^(2/3)" into a cell, the Microsoft "Excel" spreadsheet returns the error "#NUM!".
  Some calculators and programs will do the computations as expected, as displayed at right from my other graphing calculator:
  The difference has to do with the pre-programmed calculating algorithms. These algorithms generally try to do the computations in ways which require the fewest "operations", in order to process what you've entered as quickly as possible. But sometimes the fastest method isn't always the most useful, and your calculator will "choke".
  Fortunately, you can get around the problem: by splitting the numerator and denominator of the fractional power, your calculator should arrive at the correct value:
  As you can see above, it didn't matter if I first took the cube root of negative eight and then squared, or if I first squared and then cube-rooted; either way, the calculator returned the proper value of "4" الطالبات: وجدان محمد العتيبي لمى عبدالعزيز القصير R6.