Algebra: More Powers
Square Roots and More
Square Roots - review with numbers and variables
Numbers can be raised to any power in a given situation. In the section on algebraic manipulation of powers we saw that we can have any number raised to any exponent, but we assumed that the exponents were integers ... however, this is not a necessary restriction; it was just simpler to illustrate the algebraic rules in terms of integer exponenets.
In fact, you have undoubtledly encountered non-integer exponents in the past, though you might not have been aware of it at the time! The simplest non-integer power is the square root. Let us review what we do when we take the square root of a number. In taking a square root of 4 for instance, we ask ourselves what number(s) can by multiplied by themselves once in order to give 4? The answer is: well 2 × 2 gives 4 and so does (-2) × (-2). Thus, +2 or -2 raised to the power of 2 gives 4. So the square root of 4 is equal to ±2 (the notation "±" means "plus or minus"). Thus, if Sqrt is to denote the process of taking the square root we have:
22 = 4
Sqrt(22) = Sqrt(4)
2 = Sqrt (4)
and we say that the "square root of 4 is equal to 2". Same works for the value -2. If you notice above, the Sqrt process actually undoes what the raising to the power of 2 had done; in other words, in some sense this is the "opposite" process of squaring. Recall from our algebraic rules for powers that a number to a power can be raised to a power again and all we do is multiply the powers; then note that the square root process can be written as raising to the power of ½:
Sqrt(22) = (22)½ = 22 × ½ = 21 = 2
Given that many physical processes (such as the force of gravity and electricity) involve powers of 2 the square root procedure is very important to understand, and is a common fractional exponent that you will encounter in astronomy.
Numbers can be raised to fractional powers and our simplest exmple is that taking the square root of a number is equivalent to raising that number to the power of ½. Notice that the square root of a number is not always as trivial as the example above. It is not as easty to "just say" what the square root of 6.52 would be for instance. Fortunately for you, in this day and age we have calculaters and computers and you just type in "6.52" and press the [SQRT] button and get a number out. You need not worry about how the details of the general square root work. What you should get out of this section is that when you take the square root of a number you are raising it to a fraction of ½ power.
Fractional Powers - more roots
We can write fractional exponents with ageneral notation - in terms of variables. Thus, we can think of taking the square root of any positive number, x. The square root is then written as a power of one-half: x½.
The square root is an example of a fractional exponent. We can have different kinds fractional exponents, and there is no reason why they shouldn't be allowed ... you can take the so called "cube root" of a number, which involves raising to the 1/3 power. Or you can take the 4th root which involves taking a number to the 1/4 power, and so on. Here are some examples:
81/3=2 (since 2 cubed will give 8)
321/5=2 (since 2 to the 5th power gives 32)
Thus, a fractional exponent indicates that some root is to be taken. A 1/2 fraction indicates that it is a square root, and a 1/3 fraction indicates that it is a cube-root, and so on .... Generally, we can take the n-th root of numbers, and we can write this as: x1/n.
Algebraic Rules for Working with Fractional Powers
Defining the process of taking roots (square, cube, etc ...) in terms of fractional exponents has the advantage that the same basic algebraic rules discussed in the case of integer powers also hold for fractional powers. The most important of these rules to concentrate on is when we get a number raised to some power and then that power is raised to some fractional power, as in: 82/3 = (81/3)2. Note that 8 to the 1/3 power is equal to 2; then you take the stuff in the parantheses and square it (thus 2 squared) which gives 4. So: 82/3 = 4.
This is not a new rule! It just says that when a power is raised to some other power you just multiply the powers, which is the same rule we had before. It looks different because now we have allowed for fractional powers, but there's nothing strange about this. This can be written for a general case as:
These rules for powers - both integer powers and fractional powers - are extremely important to understand and be comfortable with, and as alwasy practicing problems works best. There are many many situations in astronomy where you will have to take numbers and manipulate them, and these numbers will be in equations that have powers. It is essential to be able to work with powers and take roots and understand the general procedure and rules that need to be followed.