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Rule

$na_{n} ={∣a,∣a∣, ifnis oddifnis even $

The absolute value of a number is always non-negative, so when $n$ is even, the result will always be non-negative. Consider a few example $n_{th}$ roots that can be simplified by using the formula.

$a$ | $n$ | $na_{n} $ | Is $n$ even? | Simplify |
---|---|---|---|---|

$4$ | $3$ | $34_{3} $ | $×$ | $4$ |

$-6$ | $5$ | $5(-6)_{5} $ | $×$ | $-6$ |

$2$ | $6$ | $62_{6} $ | $✓$ | $∣2∣=2$ |

$-3$ | $4$ | $4(-3)_{4} $ | $✓$ | $∣-3∣=3$ |

To write the expression for $na_{n} ,$ there are two cases to consider.

- $a≥0$
- $a<0$

Both cases will be considered one at a time.

$na_{n} =(a_{n})_{n1} $

Since both $n$ and $n1 $ are rational numbers, the Power of a Power Property for Rational Exponents can be applied to simplify the obtained expression.
$(a_{n})_{n1}$

▼

Simplify

$a$

In case of a negative value of $a,$ there are also two cases two consider.

- $n$ is even
- $n$ is odd

$a_{n_{e}}=(-a)_{n_{e}} $

Now, since $-a$ is positive, the Power of a Power Property for Rational Exponents can be applied again to simplify $na_{n_{e}} .$
$na_{n_{e}} $

▼

Simplify

NegBaseToPosPow

$(-a)_{n_{e}}=a_{n_{e}}$

$n(-a)_{n_{e}} $

RootToPowD

$na =a_{n1}$

$((-a)_{n_{e}})_{n1}$

PowPow

$(a_{m})_{n}=a_{m⋅n}$

$(-a)_{n_{e}⋅n1}$

$a⋅a1 =1$

$(-a)_{1}$

ExponentOne

$a_{1}=a$

$-a$

$n_{o}timesy⋅y⋅…⋅y =a_{n_{o}} $

Because $n_{o}$ is odd and $a$ is negative, $a_{n_{o}}$ is also negative. This means that the best candidate for $y$ is simply $a.$ $na_{n} =a $

If $a$ is non-negative, $na_{n} $ is always equal to $a.$ However, in case of negative $a,$ the value of $na_{n} $ depends on the *parity* of $n.$

$a≥0$ | $a<0$ | |
---|---|---|

Even $n$ | $na_{n} =a$ | $na_{n} =-a$ |

Odd $n$ | $na_{n} =a$ | $na_{n} =a$ |

To conclude, for odd values of $n,$ the expression $na_{n} $ is equal to $a.$ On the other hand, if $n$ is even, $na_{n} $ can be written as $∣a∣.$

$na_{n} ={∣a,∣a∣, ifnis oddifnis even $

Depending on the index of the root and the power in the radicand, simplifying $na_{m} $ may be problematic. Because real $n_{th}$ roots with an even index are defined only for non-negative numbers, the absolute value is sometimes needed.

If $n$ is even, $na_{m} $ is defined only for non-negative $a_{m}.$

Consider example $n_{th}$ roots $na_{m} $ that can be simplified by using the decision tree.

$a$ | $m$ | $n$ | $na_{m} $ | Simplify |
---|---|---|---|---|

$2$ | $8$ | $4$ | $42_{8} $ | $2_{48}=2_{2}=4$ |

$-3$ | $6$ | $2$ | $(-3)_{6} $ | $∣∣∣∣ (-3)_{26}∣∣∣∣ =∣∣∣ (-3)_{3}∣∣∣ =∣-27∣=27$ |

$x$ | $6$ | $3$ | $3x_{6} $ | $x_{36}=x_{2}$ |

$-3$ | $2$ | $8$ | $8(-3)_{2} $ | $4∣-3∣ =43 $ |

$2$ | $3$ | $6$ | $62_{3} $ | $2 $ |

$-2$ | $3$ | $9$ | $9(-2)_{3} $ | $3-2 $ |

$2$ | $6$ | $8$ | $82_{6} $ | $42_{3} =48 $ |

$-2$ | $6$ | $8$ | $8(-2)_{6} $ | $4∣-2∣_{3} =42_{3} =48 $ |

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