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

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

How many real solutions are there
to *x*^{n} = *b*?

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. |

Define
^{n}√*b*
to be this solution,
called the **real nth root of b**
(or the

^{n}√*b*=*a*;*a*^{n}=*b*.

What is the sign
of
^{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. |

How many real solutions are there
to *x*^{n} = *b*?

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 are two real solutions, one negative and one positive. |

Define
^{n}√*b*
to be the non-negative solution,
called the **principal nth root of b**
(or the

^{n}√*b*=*a*;*a*^{n}=*b*, and*a*≥ 0.

What is the sign
of
^{n}√*b*?

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. |

Ifb^{m/n}=^{n}√b^{m}.

When *b* is positive,
it's possible to define *b*^{x}
(as another positive number)
even when *x* is irrational,
but we won't pursue that in this course.
(If *b* is negative and *x* is irrational,
then *b*^{x} is imaginary.
If *b* is zero and *x* is irrational and positive,
then *b*^{x} is also zero.
If *b* is zero and *x* is irrational and negative,
then *b*^{x} is completely undefined.)

- 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*|.

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This web page was written by Toby Bartels, last edited on 2019 September 27. Toby reserves no legal rights to it.

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