If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Simplifying roots of negative numbers

Discover the magic of the imaginary unit 'i'! This lesson dives into simplifying the square root of negative numbers using 'i', the principal square root of -1. We'll explore how to rewrite negative numbers as products, and use prime factorization to simplify roots. Created by Sal Khan and Monterey Institute for Technology and Education.

Want to join the conversation?

  • starky sapling style avatar for user Wardah Zahid
    what's PRINCIPLE square root??
    (57 votes)
    Default Khan Academy avatar avatar for user
    • hopper cool style avatar for user Chuck Towle
      Wardah,
      For real numbers
      The Principal Square Root is the positive square root.
      √9 is both -3 and +3
      But the Principal square root of 9 is only 3

      For imaginary numbers,
      the Principal square root of √(-1) is i and not -i

      I hope that helps.



      It is like saying the absolute value of the square root.
      (119 votes)
  • marcimus pink style avatar for user jael.spahn
    At he said that it's impossible to use this property when both of them are negative!
    I do actually not get that point! Is there any prove? I mean it can't just be by definition, and if it is, why is it allowed to say so? I'm confused!
    (48 votes)
    Default Khan Academy avatar avatar for user
    • mr pink red style avatar for user andrewp18
      Here is the proof I posted for someone else who asked the same question a while ago:
      Great question.
      *Note that ⁺√ implies principal square root.*
      We must prove that:
      ⁺√(ab) ≠ ⁺√a • ⁺√b
      For
      a, b < 0
      If a and b are negative, then the square root of them must be imaginary:
      ⁺√a = xi
      ⁺√b = yi
      x and y must be positive (and of course real), because we are dealing with the principal square roots.
      ⁺√a • ⁺√b = xi(yi) = -xy
      -xy must be a negative real number because x and y are both positive real numbers.
      On the other hand,
      ⁺√(ab) = √[(xi)²(yi)²] = (xyi²)² = (xy)²
      Since ⁺√(ab) = (xy)² and ⁺√a • ⁺√b = -xy, our problem becomes to prove that:
      (xy)² ≠ -xy
      For
      x, y > 0
      Well this is easy! The left hand side is obviously positive and the right hand side is obviously negative, so they cannot be equal! Therefore, ⁺√(ab) ≠ ⁺√a • ⁺√b if a, b < 0. Q.E.D. Comment if you have any questions.
      (58 votes)
  • starky tree style avatar for user Joshua Herbord
    What is the difference between square root and principal square root?
    (12 votes)
    Default Khan Academy avatar avatar for user
  • leaf green style avatar for user Eric Blum
    Why can't we continue the line of reasoning Sal mentioned in the video. More specifically, why doesn't it work? I followed it along and eventually got to the conclusion that √52 = -√52, which makes no sense. If this were true, -1 would = 1, and all sorts of weird stuff would happen. What is wrong with that line of reasoning?
    (17 votes)
    Default Khan Academy avatar avatar for user
    • duskpin ultimate style avatar for user Abhishek Kumar
      Hello,
      √52 can be written as
      1) √ (+13).(+4) and
      2) √(-13).(-4)
      Now expression in (1) follows the property √a.b=√a.√b (or √b.√a)
      But in (2) expression fails to be follow this property correctly.
      Here we can go further to write
      √52=(√-13).(√-4) Now, this is where Sal says it's not right to simplify the square root.
      As √-13 can be written as √13.√-1 and similarly √-4 can be written as √4.√-1 and we know i=√-1.
      So,√52=(√13.i).(√4.i), which leads to
      √52=√13.√4.i²
      √52=√13.√4(-1) (As i²= -1)
      √52=-√52 which isn't right.
      (8 votes)
  • spunky sam red style avatar for user hchaudhry
    what is the principle square root
    (9 votes)
    Default Khan Academy avatar avatar for user
  • female robot ada style avatar for user P Deepthi sree
    when we solve a quadratic equation and in case we get the discriminant negative( i know we are gonna get complex solutions) but what do these complex solutions signify on a graph??
    (7 votes)
    Default Khan Academy avatar avatar for user
    • hopper cool style avatar for user Bruce
      That's a great question! First you need to remember what the solutions normally mean. Usually what you are trying to do is to find the x intercept. That "signifies on the graph" something you probably can understand and look at. But when the solution is complex and you are trying to think of the significance first ask yourself, how do I graph complex numbers? The answer is you need to invent a whole new concept of numbers and this thingy called the complex plane. Read this http://www.purplemath.com/modules/complex3.htm article which puts it a little differently and even has some pictures, that will solidify the concepts a little better in your brain. Complex numbers are very complex and take most people a lot of effort to understand, don't give up.
      (11 votes)
  • cacteye yellow style avatar for user infresovnas
    if i*√4*√13 is 2i*√13, then why does it is not also -2i*√13?
    (5 votes)
    Default Khan Academy avatar avatar for user
  • spunky sam blue style avatar for user Chunmun
    Are complex numbers same as imaginary numbers ?
    (5 votes)
    Default Khan Academy avatar avatar for user
    • leaf green style avatar for user ArDeeJ
      Complex numbers: all numbers of the form a + bi with real-values a and b where i is the imaginary unit.

      Imaginary numbers: all numbers of the form bi with real-valued nonzero b. A subset of the complex numbers.
      (10 votes)
  • male robot hal style avatar for user Van-Houston
    √-77 doesn't have any perfect squares in it what should I do?
    (5 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user lkshotty17
    What does he mean when he keeps saying the "principal square root"? He corrected himself multiple times in the video so he was saying that not just "square root".
    (3 votes)
    Default Khan Academy avatar avatar for user

Video transcript

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.