Question

(-i)^10

Answer 1

evaluate

Answer 2

The answer will be -1

Answer 3

yes I know but why...I get 1 when I work it out

Answer 4

do you know i^4=1?

Answer 5

yes

Answer 6

\[\frac{10}{4}=2+\frac{2}{4} \\ 10=4(2)+2 \text{ just multpied 2 on both sides } \\ i^{10}=i^{4 \cdot 2 +2} =i^{4 \cdot 2} i^2 \text{ by law of exponents } \\ i^{10}=i^{4 \cdot 2 } i^2 =(i^4)^2 i^2 \text{ also by law of exponents } \\ \text{ recall } i^4=1 \\ i^{10}=(1)^2 i^2 =i^2 \]
do you know what i^2 equals ?

Answer 7

\[i^{n}=i^{ \text{ remainder of } n \text{ divided by 4 }}\]

Answer 8

so in summary we know i^(10) will be i^2
since 10 divided by 4 gives a remainder of 2

Answer 9

and i^2=...

Answer 10

-1 @freckles

Answer 11

right

Answer 12

\[i^{10}=i^2=-1\]

Answer 13

like I undertand your work but what happens to the negative that was attached to the i

Answer 14

\[(-i)^{10}=i^{10}\]
do you know why?

Answer 15

\[(-i)^{10}=(-1 \cdot i)^{10} =(-1)^{10} i^{10} \text{ also by law of exponents } \\ \text{ but } (-1)^{10}=?\]

Answer 16

another example:
\[(-i)^{51}=(-1 \cdot i)^{51}=(-1)^{51} (i)^{51} =-1 (i)^{51}=-1 \cdot (i^{3})=-1 \cdot (i^{2} \cdot i) \\ =-1 ( -1 \cdot i) =(-1 \cdot -1)i=1i\]

Answer 17

ohhhhh is it b/c (-i)I^10 is raised to a positive root??? so the answer is 1

Answer 18

right if you had an odd power like the one in the example then you would have -1 *i^(51)

Answer 19

oh wow im stupid....I got confused b/c my teacher gave me an example with an odd exponent

Answer 20

oh

Answer 21

thank you! @freckles

Answer 22

np