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How Do You Simplify the Square Root of a Negative Number?

Simplify the square root of -60.

Summary

  1. The symbol over the -60 is the square root, or radical sign
  2. -60 has a perfect square factor of 4, so it can be simplified
  3. If we factor out a -1, then we can take the square root of 60
  4. The prime factorization of 60 is 2•2•3•5
  5. The Product Property allows you to break up the radical
  6. 'i' is the imaginary unit and equals the square root of -1
  7. Squaring and taking the square root are opposite operations, so the square root of 22 is 2!
  8. 'r' can stand for any number

Notes

    1. 'Radical' here refers to the square root symbol
    2. The square root of -60 can be rewritten as the square root of -15*4
    3. Since -60 has a perfect square factor, we can simplify this further!
    1. Factoring out the -1 will let you simplify the positive part of the radicand
    2. The radicand is the number under the square root symbol
    3. Now you can focus on the positive 60 under the radical
    4. A 'radical' is another word for the square root symbol
    1. The positive number here is 60
    2. 2, 3, and 5 are all prime numbers and cannot be factored any further
    1. 2, 3, and 5 are all prime numbers and cannot be factored any further
    2. Since there are two factors of 2, we can rewrite them as one factor with an exponent of 2
    3. Numbers to the power of 1 have an invisible exponent of 1, like 3 and 5 do here
    1. 'Radical' here refers to the square root symbol
    2. The Product Property allows you to put separate terms under their own radicals
    3. 3 and 5 are both prime, so we'll leave them together under the same radical
    1. Finally, simplify what's under each radical
    2. 'Radical' here refers to the square root symbol
    3. 'i' is the imaginary unit and is equal to the square root of -1
    4. Squaring and taking the square root are opposite operations, so the square root of 22 is 2!
    5. 3•5 = 15
    6. Rewrite everything as 2i times the square root of 15
    1. 'r' is a variable that can represent any number here
    2. Notice how you can pull out a -1 just like we did with -60 in our problem
    3. Recall that the square root of -1 is equal to the imaginary unit 'i'