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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've 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 other 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 3 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, we're 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 x2 = positive, that has two solutions, one positive and one negative. For example, x squared equals 9, it could be plus 3 or minus 3. By contrast, x2 = negative has no possible solution, x squared equals negative 9, there's no real number that satisfies that equation.

The positive and negative roots 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, x3 = positive has one positive solution and x3 = 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 0, 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 raise it to the third power and we can find the cube root of it. So, for example, the cube root of 8 is 2, the cube root of 0 is 0, and the cube root of -8 is -2.

So unlike the square root, the cube 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 10 memorized. Certainly, the first six are very important to memorize, and really, if you know all 10 it's kind of a time saver. And a couple of things I'll point out.

Notice that 8 cubed, of course, that would be 2 to the 9th. Notice that 4 cubed, that's 2 squared cubed. So that would be 2 to the 6th. It would also be, 2 to the 3rd squared or 8 squared. And of course, 8 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 tests and higher order roots, fourth root, fifth root, sixth root are even more rare. So these are not very likely topics, so 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 roots, square root, cube root, etc. We're gonna talk about all the patterns here. First of all, the square root of a, the 4th root of a, the 6th root of a, etc, these are called even roots. So when we have an even number written there, it's an even root.

And of course with a square root, there is an implied two, so that's an even root also. The cube root, the 5th root, the 7th 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 tried to take 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, it's understood that n is an integer greater than or equal to 2. For all roots, we can take the nth root of 0, and it equals 0, and the nth root of 1 equals 1, so that is true for all values of n.

All roots preserve the order of inequality. So if we have three numbers in a row, as long as they are positive numbers, we take the roots of them, as long as it's the same root we are taking of each number, then they remain in that same order of inequality. For example, suppose we had to estimate the 4th root of 50. Well, we'd wanna locate it between two fourth power.

Well, 2 to the 4th we know is 16, 3 to the 4th is 81. So because 50 is between 16 and 81, we can take the 4th root of all of those, and we see that the 4th root of 50 would have to be a decimal somewhere between 2 and 3. So the test is not gonna expect you to do anything more fancy with finding the 4th 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 1, The higher the root, the smaller the actual number. So the nth root is a higher order root than the mth root. And the mth root is smaller. So let's think about this, suppose we're taking different roots on 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, if we multiply two of the square roots together, we get 19, we have to multiply three of the cube roots together to get 19. And then we extend this logic, well, the 4th root has to be even smaller because I have to multiply four of them to get 19. I 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 1. 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 another 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 0 and 1, then things get a little bit different. And you may remember, things were different here. First of all, if we take a root, the roots are bigger than b, assuming that b is this fraction between 0 and 1. And the higher the order of 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 are taking different roots, say, of two-fifths. Well, two-fifths is less than the square root of two-fifths. This is gonna 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 1. So they get bigger and bigger and bigger, but they never get as big as 1. And this pattern continues with all higher orders of roots. So even if we had to compare two very high roots, we could say, for example, we know that 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 1. So we should be able to figure out where these four terms fall in an inequality even though we can figure up 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 1. That is, with numbers bigger than 1, taking higher order roots make it smaller and smaller, move it closer to 1.

Taking roots of numbers between 0 and 1 makes it bigger and they move up closer and closer to 1. So this is the pattern, everything gets closer to 1. And one of the reasons for that, of course, is that we can take any root of 1 and it equals 1. These properties are rarely tested, and only on the hardest quant problems on the test.

So once again, this is not something you are 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 1 equals 1, and any root of 0 equals 0. 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 1. So again, the numbers larger than 1 when we take roots, they get smaller and move closer to 1.

When we take roots of numbers that are between 0 and 1, they get bigger and they move closer to 1.

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