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I'm playing around with Raising Powers and had a strange result (it's probably not strange but I simply don't understand it).

$(-7)^2 = -49$ (wrong)

$(-5)^3 = -125$ (correct)

$(-3)^4 = -81$ (wrong)

$(-2)^5 = -32$ (correct)

My question here is, why are some of the above apparently cancelling out the negative(s) whilst others aren't?

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  • $\begingroup$ The real numbers are on a line with 0 in the middle. Multiplying a number with -1 rotates it 180 degrees. If you do that two times it makes a full turn 360 degrees. 3 times 3*180 degrees and so on. $\endgroup$ – mathreadler Mar 9 '17 at 16:29
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Remember how powers work.

$$2^5 = 2\times 2\times2\times2\times2=32$$

Also recall that $$-9\times-3=27$$ When two negatives are multiplied together, they yield a positive.

If you multiply a negative number by itself, even number of times, you will get an positive answer, but if you multiply a negative number by itself odd number of times, you will get an negative answer.

$$(-2)^3 = -2 \times -2 \times -2 = 4 \times -2 = -8$$ $$(-2)^4 = -2 \times -2 \times -2 \times -2 = 4 \times 4 = 16$$

So,

$$(-3)^4 = -3 \times -3\times -3 \times -3 = 9 \times 9 = 81$$

WHEN YOU ENTER EXPONENTS IN THE CALCULATOR, MAKE SURE YOU ADD BRACKETS AROUND THE NUMBER. We know $(-3)^4 = 81$, BUT $-3^4 = -81$

ALWAYS ADD BRACKETS.

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  • $\begingroup$ Thanks for the 'easy' answer, this made it clear for me. <br><br> Also, "Always add brackets"? Like, always, even when no negative is in play? $\endgroup$ – yokihadu Mar 9 '17 at 14:19
  • $\begingroup$ By the way you can't really break like that ^. And yeah add brackets whenever negatives are involved. In positives, the answer is always positive so its not really needed :) $\endgroup$ – K Split X Mar 9 '17 at 14:24
  • $\begingroup$ Yeah I noticed lol. Anyway, thanks once more for the clarification! $\endgroup$ – yokihadu Mar 9 '17 at 17:01
  • $\begingroup$ Np :), and btw if your satisfied with the answer you can upvote it or mark as correct (checkmark). (Im assuming your new to stackexchange) $\endgroup$ – K Split X Mar 9 '17 at 21:03
  • $\begingroup$ Hi, yeah I did so however since I'm new here the up vote isn't counted (yet). Will come back to do it properly once I have enough points to do so. $\endgroup$ – yokihadu Mar 10 '17 at 8:46
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if the Exponent is even then $$(-x)^{2m}=(-1)^{2m}x^{2m}=x^{2m}$$ if the Exponent is odd then $$(-x)^{2n+1}=(-1)^{2n+1}x^{2n+1}=-x^{2n+1}$$

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The product of 2 negative numbers is positive!

$(-7)^2 = (-7)(-7) = (-1)^2.7^2 = 49\\(-5)^3 = (-5)(-5)(-5) = (-1)^3.5^3 = (-1).125 = -125$

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  • $\begingroup$ But of course, do take note that the exponent belongs to the whole term. If you have $-3^4$ (if there are no brackets), then the exponent (4), belongs to only 3, and hence $-3^4 = -81$ is correct $\endgroup$ – Icycarus Mar 9 '17 at 13:55
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Take the example $(-1)^n$:

\begin{align} (-1)^1&=-1, \\ (-1)^2&=(-1)\cdot(-1)=1,\\ (-1)^3&=(-1)\cdot(-1)\cdot(-1)=(-1)\cdot(-1)^2=-1,\\ (-1)^4&=(-1)\cdot(-1)\cdot(-1)\cdot(-1)=(-1)\cdot(-1)^3=1,\\ &\;\;\vdots \end{align}

So whenever you rise your exponent by one, the sign changes. You see that you get $+1$ for even exponents and $-1$ for odd exponents. This generalizes to all negative bases.

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1.) If two negative terms multiplied we got positive resultant. Answer to your terms containing even powers.

2.) If one positive and one negative term multiplied result is negative term. Answer to your terms containing odd powers.

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