Raising Powers (Negative)

Question

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?

Answer 1

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.

Answer 2

if the Exponent is even then $$(-x)^{2m}=(-1)^{2m}x^{2m}=x^{2m}$$ if the Exponent is odd then $$(-x)^{2n+1}=(-1)^{2n+1}x^{2n+1}=-x^{2n+1}$$

Answer 3

The product of 2 negative numbers is positive!

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

Answer 4

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.

Answer 5

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.