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,
then the result is the same as when *x* is rational:
zero when *x* is positive,
completely undefined when *x* is negative.)

- Find
^{3}√64. - 64 = 4
^{3}, so^{3}√64 = 4. - Find
^{3}√−27. - −27 = (−3)
^{3}, so^{3}√−27 = −3. - Find √25.
- √25
means
^{2}√25, 25 = 5^{2}, and 5 ≥ 0, so √25 = 5. - Find
^{4}√−81. - −81 is negative and 4 is even,
so
^{4}√−81 is undefined (or imaginary).

- Find
−
^{3}√64. ^{3}√64 = 4, so −^{3}√64 = −4.- Find
−
^{3}√−27. ^{3}√−27 = −3, so −^{3}√−27 = 3.- Find −√25.
- √25 = 5, so −√25 = −5.
- Find
−
^{4}√−81. ^{4}√−81 is undefined (or imaginary), so −^{4}√−81 is also undefined (or imaginary).

- Find (−8)
^{2/3}. - (−8)
^{2/3}means^{3}√(−8)^{2}, (−8)^{2}= 64, and^{3}√64 = 4, so (−8)^{2/3}= 4. - Find (−27)
^{2/6}. - 2/6 = 1/3 in lowest terms,
(−27)
^{1/3}means^{3}√−27, and^{3}√−27 = −3, so (−27)^{2/6}= −3. - Find 25
^{1/2}. - 25
^{1/2}means √25, and √25 = 5, so 25^{1/2}= 5. - Find (−81)
^{3/12}. - 3/12 = 1/4 in lowest terms,
(−81)
^{1/4}means^{4}√−81, and^{4}√−81 is undefined (or imaginary), so (−81)^{3/12}is 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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