Question

Use the polar form of a negative real number, n, to explain why raising a negative number to odd powers results in negative value while raising a negative number to even powers result in positive values.

Answer 1

Answer:

Step-by-step explanation:

Let r be any positive number.Then the polar form of "-r" is r(cos(180°)+isin(180°).

Then, $(-r)^{n}=(r)^{n}(cos(n180^{{\circ}})+isin(n180^{{\circ}}))$

If n is an even integer, say n = 2k, then  

$(-r)^{n}=r^{n}(cos(k360^{{\circ}})+isin(k360^{\circ}))$

=$r^{n}(cos0+isin0)$

=$r^n$

Which is a positive number.

Thus, raising a negative number to even powers result in positive values.

Now, if n is odd integer that is n=2k+1,

$(-r)^n=r^n(cos(k360^{\circ}+180^{\circ})+sin(k360^{\circ}+180^{\circ})$

=$r^n(cos180^{\circ}+isin180^{\circ})$

=$-r^n$

which is a negative number.

Thus, raising a negative number to odd powers results in negative value.

Hence proved.