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A Simplify Negative Square Root Calculator

The negative square root is a number that is the square root of a negative number. Can you figure out how to solve the complicated equation?

First, let's ask ourselves which square roots can be simplified. To answer it, you need to take the number which is after the square root symbol and find its factors. If any of its factors are square numbers (4, 9, 16, 25, 36, 49, 64 and so on), then you can simplify the square root. Why are these numbers square? They can be respectively expressed as 2², 3², 4², 5², 6², 7² and so on. According to the square root definition, you can call them perfect squares. We've got a special tool called the factor calculator which might be very handy here. Let's take a look at some examples:You have successfully simplified your first square root! Of course, you don't have to write down all these calculations. As long as you remember that square root is equivalent to the power of one half, you can shorten them. Let's practice simplifying square roots with some other examples:(9 is a perfect square). The requirement of having at least one factor that is a perfect square is necessary to simplify the square root. At this point, you should probably know what the next step will be. You need to put this multiplication under the square root. In our example:Finally, you may ask how to simplify roots of higher orders, e.g., cube roots. In fact, the process is very analogical to the square roots, but in the case of cube roots, you have to find at least one factor that is a perfect cube, not a perfect square, i.e., 8 = 2³, 27 = 3³, 64 = 4³, 125 = 5³ and so on. Then you divide your number into two parts and put under the cube root. Let's take the following example of simplifying ³√192: Adding square roots is very similar to this. The result of adding √2 + √3 is still √2 + √3. You can't simplify it further. It is a different situation however when both square roots have the same number under the root symbol. Then we can add them just as regular numbers (or triangles). For example 3√2 + 5√2 equals 8√2. The same thing is true subtracting square roots. Let's take a look at more examples illustrating this square root property:

We're asked to simplify the principal square root of negative 52. And we're going to assume, because we have a negative 52 here inside of the radical, that this is the principal branch of the complex square root function. That we can actually put, input, negative numbers in the domain of this function. That we can actually get imaginary, or complex, results. So we can rewrite negative 52 as negative 1 times 52. So this can be rewritten as the principal square root of negative 1 times 52. And then, if we assume that this is the principal branch of the complex square root function, we can rewrite this. This is going to be equal to the square root of negative 1 times-- or I should say, the principal square root of negative 1 times the principal square root of 52. Now, I want to be very, very clear here. You can do what we just did. If we have the principal square root of the product of two things, we can rewrite that as the principal square root of each, and then we take the product. But you can only do this, or I should say, you can only do this if either both of these numbers are positive, or only one of them is negative. You cannot do this if both of these were negative. For example, you could not do this. You could not say the principal square root of 52 is equal to negative 1 times negative 52. You could do this. So far, I haven't said anything wrong. 52 is definitely negative 1 times negative 52. But then, since these are both negative, you cannot then say that this is equal to the square root of negative 1 times the square root of negative 52. In fact, I invite you to continue on this train of reasoning. You're going to get a nonsensical answer. This is not OK. You cannot do this, right over here. And the reason why you cannot do this is that this property does not work when both of these numbers are negative. Now with that said, we can do it if only one of them are negative or both of them are positive, obviously. Now, the principal square root of negative 1, if we're talking about the principal branch of the complex square root function, is i. So this right over here does simplify to i. And then let's think if we can simplify the square root of 52 any. And to do that, we can think about its prime factorization, see if we have any perfect squares sitting in there. So 52 is 2 times 26, and 26 is 2 times 13. So we have 2 times 2 there, or 4 there, which is a perfect square. So we can rewrite this as equal to-- Well, we have our i, now. The principal square root of negative 1 is i. The other square root of negative 1 is negative i. But the principal square root of negative 1 is i. And then we're going to multiply that times the square root of 4 times 13. And this is going to be equal to i times the square root of 4. i times the square root of 4, or the principal square root of 4 times the principal square root of 13. The principal square root of 4 is 2. So this all simplifies, and we can switch the order, over here. This is equal to 2 times the square root of 13. 2 times the principal square root of 13, I should say, times i. And I just switched around the order. It makes it a little bit easier to read if I put the i after the numbers over here. But I'm just multiplying i times 2 times the square root of 13. That's the same thing as multiplying 2 times the principal square root of 13 times i. And I think this is about as simplified as we can get here. (Source: www.khanacademy.org)

The first use of the square root symbol √ didn't include the horizontal "bar" over the numbers inside the square root (or radical) symbol, √‾. The "bar" is known as a vinculum in Latin, meaning bond. Although the radical symbol with vinculum is now in everyday use, we usually omit this overline in the many texts, like in articles on the internet. The notation of the higher degrees of a root has been suggested by Albert Girard who placed the degree index within the opening of the radical sign, e.g., ³√ or â´√.

The last question is why is the square root operation called root regardless of its true origin? The explanation should become more evident if we write the equation x = â¿√a in a different form: xâ¿ = a. x is called a root or radical because it is the hidden base of a. Thus, the word radical doesn't mean far-reaching or extreme, but instead foundational, reaching the root cause. (Source: www.omnicalculator.com)

If you're looking for the square root graph or square root function properties, head directly to the appropriate section (just click the links above!). There, we explain what is the derivative of a square root using a fundamental square root definition; we also elaborate on how to calculate square roots of exponents or square roots of fractions. Finally, if you are persistent enough, you will find out that square root of a negative number is, in fact, possible. In that way, we introduce complex numbers which find broad applications in physics and mathematics.

The first use of the square root symbol √ didn't include the horizontal "bar" over the numbers inside the square root (or radical) symbol, √‾. The "bar" is known as a vinculum in Latin, meaning bond. Although the radical symbol with vinculum is now in everyday use, we usually omit this overline in the many texts, like in articles on the internet. The notation of the higher degrees of a root has been suggested by Albert Girard who placed the degree index within the opening of the radical sign, e.g., ³√ or â´√. (Source: www.omnicalculator.com)