10
Time to talk about our roots. Fundamental definition. Given a positive number y, the n-th root of y means the unique positive number x having the property that x to the power n equals y. The usual notation for this concept uses that symbol there called the radical. And radical, in fact, is a synonym for a n-th root.
42.8
To summarise, to say it once again, given two positive numbers to say that x is the n-th root of y is equivalent to saying that y is equal x to the power n. Now when n equals 2, we usually don’t write the 2 with a radical sign. We just write square root of y as indicated. Some comments. Every positive number has a unique n-th root. Now, for example, it’s true that there are two numbers whose square is equal to 4– the numbers 2 and minus 2. But the square root of 4 is 2. It’s not minus 2. Radicals are defined only for positive numbers.
85.7
We could try and define the cubic root of minus 8 as minus 2 for example because after all, it is true that minus 2 to the power 3 equals minus 8. But this is not a good idea for reasons that shall be clarified later. So we assign no meaning to this symbol. Positive numbers only for radicals. Let’s look at the graph of the function square root of t. t, therefore, is restricted to r plus 0 infinity. The graph turns out to look like this. It rather looks like a half of a parabola that’s lying on its side. And in fact, it is. Facts about radicals?
133.2
Well, if you have two positive numbers a and b, the n-th root of the product a,b is the product of the n-th roots. We can prove something like that fairly easily. First of all, we notice that both the n-th roots are positive by definition, and therefore, so is their product. All we need to show now is that when you take the power n of that number, you get a,b. That will, by uniqueness, that will prove the formula we wish to establish. When you take the power n, you– by the laws of exponents that we’ve already memorised with our million examples– we get exactly a b, which proves the theorem, allowing us to say box. Other facts about radicals?
180.3
Well, there’s a bunch of them. For example, this one. And look at this new one up here. It says that if you’re taking the m n-th root of the m power of a, you can cancel out the m under certain circumstances and get simply the n-th root of a. Bear in mind that a has to be positive for this to be true. Another law is this one. And finally, we have that one. All these rules have to be known, and it’s rather easy to get to know them with a little practise. But even after you know them, you must exercise some caution, experience indicates, in applying them. For example, what is the square root of x squared?
228.6
We’re very tempted to say that it’s equal x. But in fact, that’s not true in general. The square root of minus 3 squared– square root of 9– is 3, and not minus 3. So that expression is wrong in general unless we specify that x is positive, or unless we put absolute values in because we’re using this law, which as you know, we derived only for positive a. The real true formula is that the square root of x squared is the absolute value of x for any real number x. Similarly, we can prove this formula, which I leave as an exercise. Bear in mind, the absolute a that occurs in it.
276.2
Another caution that I wish to make is that we’re often very tempted to invent a new rule because it would be so convenient in a calculation. It is not true that the square root of a plus b is the square root of a plus the square root of b. Here’s a simple example. The square root of 9 plus 16 is 5, but root 9 plus root 16 is 7. Remember in mathematics, we’re not allowed to invent the rules. We don’t even get a vote. Nobody does.
Definition and fundamental properties of root functions