Imaginary roots of negative numbers | Imaginary and complex numbers | Precalculus | Khan Academy

Main ideas / concepts

• The goal is to simplify the principal square root of a negative number: [ \sqrt{-52} ]

• It is assumed the principal branch of the complex square root function is being used, so negative inputs are allowed and the result may be imaginary/complex.

• A key algebraic rule discussed:
  • You can split (\sqrt{ab}) into (\sqrt{a}\sqrt{b}) only if:
    • both (a) and (b) are positive, or
    • exactly one of them is negative.
  • You cannot do it when both (a) and (b) are negative, since that can lead to incorrect results.

Step-by-step methodology used in the video

1. Start with: [ \sqrt{-52} ]

2. Rewrite (-52) as a product involving (-1): [ -52 = (-1)\cdot 52 ] so [ \sqrt{-52}=\sqrt{(-1)\cdot 52} ]

3. Apply the product rule for principal square roots (valid here because one factor is negative and the other is positive): [ \sqrt{(-1)\cdot 52}=\sqrt{-1}\cdot \sqrt{52} ]

4. Substitute the principal square root of (-1): [ \sqrt{-1}\ \text{(principal)} = i ]

5. Simplify (\sqrt{52}) using prime factorization: [ 52 = 2\cdot 26 = 2\cdot 2\cdot 13 = 4\cdot 13 ] [ \sqrt{52}=\sqrt{4\cdot 13}=\sqrt{4}\cdot \sqrt{13}=2\sqrt{13} ]

6. Combine with the factor (i): [ \sqrt{-52}=i\cdot (2\sqrt{13})=2\sqrt{13}\,i ]

Final simplified form: [ \boxed{\sqrt{-52}=2\sqrt{13}\,i} ]

Speakers / sources featured

• Khan Academy (video narrator/instructor) — the speaker is not explicitly named in the provided subtitles.