Numbers, Variables, and Rules
Powers: Definition and Simple Powers with Numbers
Even though you may not be aware of it, you have already encountered the concept of "raising a number to a power". Consider the following simple example: 3 × 3 = 9. We can re-write this as: 3(2)=9. This notations simply means that if we take the number 3 and multiply it by itself we get 9. Let us consider: 3 × 3 × 3 = 3(3) = 27. This means that three threes multiplied together gives us 27.
In the above examples we say that the number 3 is raised to a power. If 3 is raised to the 2nd power, we get 9 and if 3 is raised to the 3rd power we get 27. In fact, writing the number of times that we multiply something into itself as a power - with the notation of the power as being a superscript to the right of the number - can save us quite a lot of space!. Suppose we were multiplying 3 by itself 6 times, and compare the following:
3x3x3x3x3x3 = 729
36 = 729
Clearly, the 2nd way of writing has the advantages of: (i) being more efficient and (ii) being easier to see how many times the number is being multiplied into itself. Futhermore, as you will see below, this way of indicating numbers and powers makes the algebraic manipulation of numbers and multiplication of large values much much easier.
Why are Powers Relevant?
Powers - or rather, the exponential notation introduced above - is extremely important in astronomy. The very basic reason for this is that astronomy forces us to expand our understanding of scales and sizes of objects. You may go back and review the sections on powers of 10 to see what kind of scales we are talking about when we refer to concepts such as "the age of the earth", "the age of the universe", "the size of the sun", etc .... Let us take the Earth-Sun distance for instance; you can look this up in any astronomy textbook and you will notice a number written as follows: 150 × 106 km. This means that this distance is 150 times a million km (note: 106 means 10 mulitiplied by itself 6 times, which is a million). If you had to write this number out you would write: 150,000,000. Now imagine trying to use these number in mathematical relationships to calculate some other value; there is a simple clear advantage to the exponential power notation!
There are many occasions also that one is simply interesting in orders of magnitude, and the exact details of a number are not important. The exponential power schema or writing out numbers provides a simple way of expressing the "degree of largness/smalness of an object" without giving details.
Powers with Variables
To generalize the notion of powers fruther, we can imagine any number being multiplied by itself any number of times. We call a general - underspecified - number a variable, and the canonical notation for an algebraic variable is x. Thus, we say that x2 is "x raised to the power of two" or equivalently "x squared" (which is x mulitplied by itself).
In fact, if we multiply the variable x by itself a number of times (say n - times) we get: xn. The number represented here as x is called the base, and n is called the power or exponent. The basic definition of a number written in exponential notation states that the base should be multiplied by itself the number of times indicated by the exponent.
52 = 5 × 5
b4 = b × b × b × b.
Algebraic Rules for Working with Powers
From the above definition of powers some basic algebraic rules follow. Consider the following; suppose we multiply the following:
Notice that in multiplying the 2 to the 3rd power by 2 to the 2nd power we get number that is 2 to the fifth power. Thus, in multiplying numbers in exponential notation - following the rules of number multiplication - the power must add: 23 × 22 = 2 3 + 2 = 25. This, in fact, generalizes to any base and any power, and we can write it interms of our variable notation introduced above.
From this, the division rule for powers is easy to see as well:
Thus, instead of adding we subtract powers when carrying out a division: 23 / 22 = 2 3 - 2 = 21.
Another rule that emerges out of the algebra of exponential powers is that if a number raised to a power and is in the denominator of a fraction, then it can be written as "raised to a negative power". For instance, consider the following example:
1/22 = 2-2 = 1/4.
Another important observation about exponential powers rests in the fact that a number raised to some power can itself be raised to a power again! An example can best illustrate this point. Consider:
(23) × (23) = 23+3 = 26
The number on the left side of the equations sring is the quantity "2 cubed" multiplied by itself. According to our definition above - of algebraic powers -we can write this as: (23)2 ... that is, 2 cubed raised to the power of two. Thus, when a number to some power is raised to a power again, we multiply the two powers to get the resultant value:
(23)2 = 2(3) × (2) = 26.
Finally, one more important piece of knowledge about exponents: a number raised to the power of zero is alwasy equal to 1. Thus: 20 = 130= -40 = 1.
These are the basic rules for working with and manipulating exponents and powers. You need to keep these in mind and they will show up fequently in making qualitative and quantitave argumetns in astronomy. Here is a summary of what has been mentioned above, in more general notation.
Summary of Algebraic Rules for Powers
This should give you a basic start on working with and understanding powers and exponents. Remember that in the general notation used for the above rules we write x to mean that for any given number the rule works ... i.e., you can put any number in place of x and the above rules should hold. Same goes with the exponents n, m. We write it in this general notation because it is simpler to state rules this way. Do not let this notation with letter variables confuse you. When you encounter a problem and you see that you need to take say 5 to the third power and multiply it by 5 to the 10th power, you simply realize that for the given situation: x = 5, and n=3 and m=10 .... Practicing with problems and applying these rules will make them easier to understand.