If a
is negative: |
If a is zero: |
If a is positive: |

a^{n} is negative; |
a^{n} is zero; |
a^{n} is positive. |

If b
is negative: |
If b is zero: |
If b is positive: |

there's one real solution, which is negative; |
there's one real solution, which is zero; |
there's one real solution, which is positive. |

^{n}√*b*=*a*;*a*^{n}=*b*.

If b
is negative: |
If b is zero: |
If b is positive: |

^{n}√b
is negative; |
^{n}√b
is zero; |
^{n}√b
is positive. |

If a
is negative: |
If a is zero: |
If a is positive: |

a^{n} is positive; |
a^{n} is zero; |
a^{n} is positive. |

If b
is negative: |
If b is zero: |
If b is positive: |

there's no real solution; |
there's one real solution, which is zero. |
there's two real solutions, one negative and one positive. |

^{n}√*b*=*a*;*a*^{n}=*b*, and*a*≥ 0.

If b
is negative: |
If b is zero: |
If b is positive: |

^{n}√b
is undefined (or imaginary); |
^{n}√b
is zero; |
^{n}√b
is positive. |

- Find
^{3}√8. - 8 = 2
^{3}, so^{3}√8 = 2. - Find
^{3}√−27. - −27 = (−3)
^{3}, so^{3}√−27 = −3. - Find √16,
meaning
^{2}√16. - 16 = 4
^{2}and 4 ≥ 0, so √16 = 4. - Find
^{4}√−81. - −81 is negative and 4 is even,
so
^{4}√−81 is undefined (or imaginary). - Find
−
^{3}√8. ^{3}√8 = 2, so −^{3}√8 = −2.- Find
−
^{3}√−27. ^{3}√−27 = −3, so −^{3}√−27 = 3.- Find −√16.
- √16 = 4, so −√16 = −4.
- Find
−
^{4}√−81. ^{4}√−81 is undefined (or imaginary), so −^{4}√−81 is also undefined (or imaginary).- Find
^{3}√*x*^{3}. ^{3}√*x*^{3}=*x*.- Find
^{3}√−*x*^{6}. - −
*x*^{6}= (−*x*^{2})^{3}, so^{3}√−*x*^{6}= −*x*^{2}. - Find
√
*x*^{2}. *x*^{2}= (*x*)^{2}and*x*^{2}= (−*x*)^{2}; either way,*x*^{2}= |*x*|^{2}and |*x*| ≥ 0, so √*x*^{2}= |*x*|.- Find
^{4}√16*x*^{8}*y*^{4}. - 16
*x*^{8}*y*^{4}= (2*x*^{2}|*y*|)^{4}and 2*x*^{2}|*y*| ≥ 0, so^{4}√16*x*^{8}*y*^{4}= 2*x*^{2}|*y*|.

Ifb^{m/n}=^{n}√b^{m}.

Go back to the course homepage.

This web page was written between 2010 and 2016 by Toby Bartels, last edited on 2016 July 14. Toby reserves no legal rights to it. The permanent URI of this web page is

`http://tobybartels.name/MATH-1100/2016SU/roots/`

.