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Other Roots

In this lesson, we’re going to talk about other roots. Cube roots, fourth roots, this sort of thing. So, just as the square root more or less, quote, undoes the act of squaring. Of course, we learned that that’s not exactly true. There are higher order roots that similarly undo the other powers. For example, cube roots undo the act of cubing something.

So just as we can raise something to the third power, we can find what number raised to the third power would equal a certain result. So first of all, let’s talk about the notation. Square root of k, radical k denotes the square root of k. The notation for all other roots is similar. We use the same radical symbol.

But for all of the roots we put a small number in front of the radical to denote the order of the root. In other words, are we taking a cube root, a fourth root, a fifth root, etc. So for a cube root, what we would do is put a little three in front of the radical sign, and that would denote the cube root of k. We also have to think about positive and negative, cuz positive and negative numbers get kind of tricky here.

Remember that, first of all with squaring, a positive squared is positive and a negative squared is also positive, okay? So this is true for squaring. Therefore it’s true that if we have x squared equals a positive, that has two solutions, one positive and one negative. For example, x squared equals nine, it could be plus three or minus three.

By contrast, x squared equals a negative has no possible solution. X squared equals negative 9, there’s no real number that satisfies that equation. The positive and negative rules for cubes are a little bit different. Remember that if we cube a positive, we get a positive, but if we cube a negative, we get a negative.

Therefore, x cubed equals a positive has one positive solution, and x cubed equals a negative has one negative solution. So you don’t get the problem of double roots in one case and no solution in another case, as we got for squaring. With square roots, we can find the square root only of positives.

Of course, we could also find the square root of zero, but we cannot take the square root of a negative, okay? So for much the same reason, just as we can’t square something and get a negative, we can’t take the square root of a negative. But the rules are very different for cube roots. We can take the cube root of any number on the number line, positive, zero, or negative.

So every single number on the negative line we can, we can raise it to the third power and we can find the cube root of it. So for example, the cube root of eight is two. The cube root of zero is zero and the cube root of negative eight is negative two. So unlike the square root, the cubed root can have a negative output. When we put in a negative we get out a negative.

For quick computations, it’s good to have the cubes up to ten memorized. Certainly the first six are very important to memorize, and really if you know all ten, it’s kind of a time saver. And a couple of things I’ll point out. Notice that eight cubed, of course that would be two to the ninth. Notice that four cube, that’s two squared cube, so that would be two to the sixth, it would also be two to the third squared, or eight squared.

Of course eight squared is 64 also, so there are some patterns here that we can observe practicing our laws of exponents. So if we have these cubes memorized, automatically we have a bunch of cube roots memorized. And so that can be very convenient. Cube roots are relatively infrequent on the test.

And higher order roots, fourth root, fifth root, sixth root are even more rare. So, these are not very likely topics. I’ll warn you right now. But I’ll say a few things in general about all roots. So this is true for every possible root, square root, cube root, et cetera, et cetera.

We’re gonna talk about all the patterns here. First of all, the square root of a, the fourth root of a, the sixth root of a, et cetera, these are called even roots. So when we have an even number written there, it’s an even root. And of course with the square root, there’s an implied two, so that’s an even root also.

The cubed root, the fifth root, the seventh root, these are called odd roots. So we’re distinguishing all the even roots from all the odd roots. And the reason we’re doing this is because the same positive and negative thing we’ve talked about with squares and cubes extends to all the evens and odds. As with square roots, we can take any even root of a positive number, which results in a positive output.

But we cannot take an even root of a negative number. We try to take the, an even root of a negative number, it is undefined. It does not equal anything on the number line. By contrast, as with cube roots, any odd root of a positive is positive and any odd root of a negative is negative. So it follows that same plus and minus pattern.

For other properties, I will use the notation with an n before the radical sign, so the nth root of a. And here I’m talking about a general root, and of course we’re, it’s understood that n is an integer greater than or equal to two. For all roots, we can take the nth root of zero and it equals zero and the nth root of one equals one.

So that is true for all values of n. All roots preserve the order of inequalities. So if we have three numbers in a row, as long as they’re positive numbers, we take the, the roots of them, as long as it’s the same root we’re taking of each number, then they remain in that same order of inequality. For example, suppose we had to estimate the fourth root of 50.

Well, we’d want to locate it between two fourth powers. Well, two to the fourth we know is 16. Three to the fourth is 81. So, because 50 is between 16 and 81 we can take the fourth root of all of those, and we see that the fourth root of 50 would have to be a decimal somewhere between two and three.

So, the test is not going to expect you to do anything more fancy with finding the fourth root of 50, but as long as you can figure out which two integers it’s between, that is fine. We can also compare the size of different order roots. So for a number greater than one, the higher the root, the smaller the actual number.

So the nth root is a higher order root than the mth root, and the nth root is smaller. So let’s think about this. Suppose we’re taking different roots of 19. Well of course the square root of 19 has to be smaller than 19. Now the cube root of 19 has to be even smaller because it only takes two of the square root, we multiply two of the square roots together, we get 19.

We’d have to multiply three of the cube roots together to get 19. And then we extend this logic, well the fourth root has to be even smaller because I’d have to multiply four of them to get 19. I’d have to multiply five of these, or six of these and so forth. So it means that as we get higher and higher order roots, all of these numbers get smaller, but they’re always larger than one.

Now it must be true, even if we can’t figure out what the decimals are, it absolutely must be true that the 30th root of 19 is less than the 20th root of 19. So in other words, we should be able to make that comparison, even though we can’t figure out the exact values of those decimals. Now when we get into that region between one, zero and one, then things get a little bit different.

And you may remember things were different here. First of all, we take a root. The number, the roots are bigger than b, assuming that b is this fraction between zero and one. And the higher the order the root, the higher it gets. So n is a higher order root.

It’s higher, the nth root of b is higher than the mth root of b. So suppose we’re taking different roots, say, of two-fifths. Well, two-fifths is less than the square root of two-fifths. This is going to be less than the cube root, which is less than the fourth root, and so forth.

We continue this pattern. It turns out that all these roots remain less than one. So they get bigger and bigger and bigger, but they never get as big as one. And this pattern continues with all higher order of roots. So even if we had to compare two very high roots we could say for example we’d know that the, the 50th root of two-fifths that has to be greater than two-fifths, but it has to be less than the 75th root of two-fifths.

And the 75th root of two-fifths still has to be less than one. So we should still be able to figure out where these four terms fall in an inequality even though we can’t figure out the exact decimal values of those roots. One way to summarize this, we could say, the higher the order of the root, the closer the result is to one.

That is, with numbers bigger than one, taking higher order roots, make it smaller and smaller, and move it closer to one. Taking root of numbers between zero and one makes it bigger, and they move up closer and closer to one. So this is the pattern. Everything gets closer to one.

And one of the reasons for that, of course, is that we can take any root of one and it equals one. These properties are rarely tested and only on the hardest Quant problems on the test. So once again, this is not something you’re gonna see every time you sit down for the test.

These are very rare problems. In summary, unlike with square roots, we can take cube roots of both positive and negatives. That’s a big idea. In fact, we can take any even root of positives only, not negatives, but we can take any odd root of any number on the number line.

That’s also a really big idea. Any root of one equals one, any root of zero equals zero. All roots preserve the order of inequalities, assuming all the numbers are positive. And the higher the order of a root, the closer the result is to one. So again, the numbers larger than one, when we take roots, they get smaller, move closer to one.

When we take roots of numbers that are between zero and one, they get bigger, and they move closer to one.