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Square Roots


Now we can talk about square roots. In many contexts in math, we know from the problems square of a number or we can do a calculation and get the value of the square of the number. And then we have to solve for the original number that was squared. So, for example, we might wind up with a situation such as x squared = 5. Now if the square of the desired number happens to be a perfect square, then that's great, then that's easy to solve.

Getting x squared equal to a perfect square is a luxury. And that doesn't usually happen. If you consider all the numbers say from 1 to 100, 1 to 100 inclusive only 10 of those are perfect square the other 90 are not perfect squares. And perfect squares become more and more rare as numbers get bigger. Square roots and square notation give us a way to talk about the situation in which the square of a variable does not equal a perfect square.

Again, this is much more common than equalling a perfect square. So for example consider x squared = 2. Well of course 2 is not a perfect square. There is no integer we could square to get 2. And suppose we wanna solve for x, well of course x, it could have two values, there could be something negative that we square to get the positive, or there could be something positive we square to get the value.

But how would we talk about that value? What number squared would equal 2? Well, again, clearly it's not an integer. It would have to be a decimal and that decimal would have to be between 1 and 2 because 1 squared is 1 and 2 squared is 4. So to get something squared equalling 2, it have to be between those two.

So we denote the positive number that, when squared, would equal two as this. And we can call that, sometimes that's called square root of 2, sometimes that's just called root 2, or radical 2, different ways to talk about this number. The square root and the act of squaring undo each other. Precisely the way that addition and subtraction undo each other. Precisely the way that multiplication and division undo each other.

They are inverse operations or from a more sophisticated point of view, they are inverse functions. And so this number, square root of 2, that's actually a real bona-fide number. It has a right to exist, it has the same right as 5 or 13 or any other real number on the number line. Square root of 2 is bona-fide number, it lives on the number line and in fact it's the decimal that leaves somewhere between 1 and 2.

The way that this number, square root of 2, is connected to the rest of mathematics is by virtue of the fact that we know, by definition, if we were to square it, we would get 2 as a result. We mentioned that the equation x squared = 2 would actually have two solutions, one positive and one negative. When that sign is written, the square root sign is written, it means the positive root only.

If we wanted to talk about the negative root, we would have to actually write in a negative sign and say negative square root of 2. And so that's the way that we could talk about both the positive and the negative. Now many students get confused about this issue, when to include or not to include the negative square root. And this is a very subtle distinction.

Situation number one, if the square root sign is written by the test-maker, if it's actually printed on the page as part of the problem, then that means consider the positive roots only. But many times what happens in a problem, there's a variable squared. Or in other words, we do our calculations, rearrange our equations, we wind up with a variable squared.

And then we ourselves have to initiate the process of taking a square root in order to solve for x. Well that happens if we're the ones initiating taking a square root, then we always have to consider both the positive and the negative root. So here are two simplistic practice questions, obviously these are not the format of actual test questions, these are kind of cook book easing.

But pause the video, and think about this, and think about whether these two questions have the same answer or not. Okay, question number 1 says is root 2 > 1? Well, notice that root 2 is printed on the page. It is actually part of the problem. And that symbol, that square root symbol, means take the positive root only.

And so we're talking about only the positive square root of 2. Is that bigger than 1? Well, of course it's bigger than 1, it's between 1 and 2. So yes, it's bigger than 1. Number 2, very, very different, now we're talking about a variable k, k squared = 2. We wanna know something about k, so we ourselves have to initiate the action of taking the square root.

Well, when we take the square root to solve for k, it could mean that k is the positive square root of 2 or it could mean that k is the negative square root of 2. We don't know whether k is positive or a negative. Well, if it's a positive square root of 2 it is bigger than 1, but if it's a negative square root of 2, it's not bigger than 1.

Because every positive is bigger than any negative. And so we can't determine it, we don't know, it can't be determined, so C is the answer here. And so it's very important to understand that these are two very different questions that are answered in very different ways. Now I'll point out, of course, when we're taking the square root of a positive number the output is always positive.

Okay, so if q is positive then square root of q is positive. So this brings up the question, what happens if q is not positive? So first of all, can we take the square root of zero? And the answer is, of course, we can, absolutely. We know that 0 squared is 0, and so if 0 squared is 0, it means that the square root of 0 is 0.

We can take the square root of 0 and it equals 0, perfectly fine, no ambiguity. Can we take the square root of a negative? Hmm, now we're on tricky ground. Technically the answer is yes, but we have to be careful here. As you may recall, the square root of a negative gives us an imaginary number. And so what this means is, if we're taking the square root of a negative, we're leaving the real number line.

We're not gonna have any answer value, that is on the real number line, but it can be done and in fact what happens is square root of 25, what that equals is 5i. And so this is an imaginary number. And so, if the problem is specifying that all the numbers are real numbers, then this is not gonna be a possibility cuz 5i is not a real number.

It does not live on the real number line, it lives somewhere else. We will talk about imaginary numbers and complex numbers in more detail later. For the rest of this video, we're just gonna be considering real numbers and not imaginary numbers, but that is coming up later. But just in summary, I'll say, if A is > 0 or = 0, then the square root of A is also > 0 or = 0.

If A is < 0, then square root of A is a multiple of i, we've left the real number line. Notice that square root of A never has an output of a negative real number, and that's important. So for example, if the test writes square root of A = B, and we're told A and B are real numbers, then by virtue of writing that with an equal sign, the test is guaranteeing that this is a sensible and completely valid mathematical equation.

And so what this means is that, we know immediately that A has to be greater than or equal to 0 and b has to be greater than or equal to 0. So technically, that sign, the square root sign, is called the principal square root sign. Which means that of the two possible square roots, this particular sign returns the positive root only.

Many people don't know that. Many people just call it the square root sign and they don't realize it's actually a much more subtle technical thing. It's the principal square root sign, which means the positive only square root sign. So if the test says, solve the equation x squared = 5, then of course, there are two solutions.

The positive solution is x = root 5, the negative solution is x =- root 5. Those are the two numbers on the number line that when squared equals 5, they're symmetrical around zero, of course. And by definition, when we square them we get 5. Now the test will expect you to recognize and simplify square roots of perfect squares.

So for example, if it gives you square root of 36, it's gonna expect you to know that that = 6. But it's very important, the test will not expect you to come up with the exact value of a square root that doesn't come out evenly. So if a square root of 41 shows up, nobody on earth expects you to do square root of 41 in your head.

So you don't have to worry about that. But the test will expect you to know some basic approximations. Now, of course, you could find the decimal value of square root of 41 with a calculator, but often the test isn't even interested in that. It's interested in your ability to approximate. So first of all for positive numbers, the square root preserves the order of inequality.

What does that mean? So, in other words, if we know that A < B < C, if they are ranked in that order, the actual numbers themselves, then when we take the square root of all of them, they're still ranked in that same order. Taking the square root does not change the order of an inequality. Again, as long as we're guaranteed that all the numbers are positive, this is true.

So what that means is we know that 41 it is between 36 and 49. It's between those two perfect squares. So if we take a square root of those three numbers that tells us that the square root of 41 has to be between 6 and 7. And so that's the kind of approximating that the test would expect you to be able to do.

And you should be able to figure out relatively quickly for any square root that doesn't come out evenly, at least it would be between which two integers. You could approximately locate it on the number line. So again, you are not expected to know that exact value, but you must know the integers that it's between. There are also a few simple square roots for which you should know rough approximations.

And these are the three big ones to know, square root of 2 is 1.4 approximately, square root of 3 is 1.7 approximately. Square root of 5 is 2.2 approximately, if you know those three, and just know them to in approximations of a tenth. Now obviously, these are decimals that go on forever and ever, you don't have to worry about that.

If you just remember them to the value of a tenth then you can do a great deal of approximating, as we will see in other videos. So that sign, the principle square root sign is an operator, just like add, subtract, multiply and divide, and like those, it can operate on numbers or variables. We have to be mindful with this operator, it undoes squaring, but it is not an exact opposite.

For example, suppose y is negative, then of course y squared would be positive. Then the square root of y squared would also be positive. And so, in other words, we start with a negative, but if we square and square root we don't end with a negative, we end with a positive. So we don't wind up back where we started. Technically, because it's always positive whether y is positive or negative, we have to say that the square root of y squared = the absolute value of y.

In other words, it's something that's always positive. Well, if it's zero, of course it equals zero, but otherwise it's positive. If a variable squared appears in the problem, and we want to take the square root and undo the square, again, we have to include the positive and negative sign to account for both roots. So if we get x squared = K, then we know that x = plus or minus the square root of K.

Here's a practice problem. Pause the video and then we'll talk about this. Okay, so let's talk about this. If x- 3 squared = 16, it means that we can take a square root of both sides. And, of course, because we're taking the square root ourselves, we have to include the plus or minus sign.

So x-3 = plus or minus 4. And then we get two equations, one where x-3 = positive 4, another where x-3 = -4 And so we can solve and get either x = 7 or x = -1, and those are the two solutions to that original equation. Let's check this, let's just make sure.

If x equals 7 then 7 minus 3 is 4, or squared is 16. If x equals negative 1, negative 1 minus 3 is negative 4. And negative 4 squared also equals 16. So, those are two solutions to that equation. We also have to consider how square roots relate to the topic of exponential growth, which we discussed earlier in this module.

There we said if 1 < b, then positive powers of b get bigger. And so if we're talking about a number bigger than 1, raising it to a power makes it bigger and bigger. In particular if b is bigger than 1, then the square of b has to be bigger than b. And of course, if b equals 1, then 1 squared equals 1, and then that's the only time that would be equal.

If squaring makes these numbers bigger, then taking a square root must make these numbers smaller. And so again, if we're talking about numbers bigger than one, if we square it, we get a bigger number. If we take a square root, we get a smaller number. So all that is true for numbers bigger than one.

But as we saw in that lesson, everything is opposite for positive numbers less than one, in other words, the fractions between 1 and 0. If b is a fraction between 1 and 0, then when we raise powers of it, when we raise it to powers, it actually gets smaller and smaller and smaller. So three to a power, keep on increasing the power, three gets bigger and bigger and bigger.

But one-third to a power keeps on getting smaller and smaller and smaller, each time we're taking one-third of what we had previously. In particular, if we have a fraction between 0 and 1, then when we square it, we get a smaller number. So if b is a fraction between 0 and 1, b > b squared. Well if squaring makes these numbers smaller, then it follows that taking the square root going the opposite direction must make those numbers bigger.

And so again, for a fraction between 0 and 1, the square root of b would have to be > b. Let's think about this, think about the number, we'll start with two-thirds. Is the fraction greater than one half? Now if we square it, of course what that means is two-thirds x two- thirds, we multiply across on the numerator and the denominator, we get four-ninths.

And four-ninths that's a fraction less than one half. And so it's clear that the square root of four-ninths = two-thirds. And two-thirds > four-ninths, because four-ninths is less than one half and two-thirds is greater than one half. And so this is one numerical example of how taking the square root of a fraction leads to a bigger number.

Here's a practice problem, pause the video and then we'll talk about this. Okay let's think about this. If 7K is a positive integer, and if square root of K > K, then is K an integer? Well if square root of K > K, then K must be a number between 0 and 1, it can't possibly be an integer.

It would have to be either one-seventh or some multiple of one-seventh. Two-sevenths, there's three-sevenths, or so forth, so we absolutely know that it's the case that it cannot be an integer. And so the answer is B, there's nothing uncertain here, it absolutely cannot be an integer. In summary, that sign, that very tricky sign, most people call it the square root sign.

Really, it is the principal square root sign. The principal square root sign always has a positive output. If we take the square root ourselves We must consider both the plus and the minus root. And so again, very important if the problem prints the square root sign, then we're dealing with a positive root only.

If the problem has a variable squared or if in our own work, in our own algebra, we wind up with a variable squared and we, ourselves, have to take the square root, well then we have to consider both the positive and the negative. We covered some basic approximations for square roots, it's important to know those approximations. We talked about if b is a number > 1, then, of course, squaring makes the number bigger, but taking the square root makes the number smaller.

And if b is a fraction between 0 and 1, then squaring it makes it smaller and taking the square root makes it bigger.

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