Question

please help me in evaluating (-1)^(1/4)

Answer 1

it is nothing but 4th root of -1

Answer 2

if you are working in complex numbers there are 4 of them. if you are working in real numbers there are none

Answer 3

yes but while solving it gives two answers i and 1 how is it possible in mathematics

Answer 4

ok i am not sure what you are doing exactly, but 1 is certainly not a solution because
\[1^4=1\] not
\[1^4=-1\]

Answer 5

if you want you can write
\[-1=e^{\pi i}\] and then
\[(-1)^{\frac{1}{4}}=(e^{\pi i})^{\frac{1}{4}}=e^{\frac{\pi}{4}i}\]
or if you cannot do this you can write
\[-1=\cos(\pi)+i\sin(\pi)\] and get
\[(-1)^{-\frac{1}{4}}=\cos(\frac{\pi}{4})+i\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i\]

Answer 6

the other solutions to
\[x^4=-1\] are
\[\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i\]
\[-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i\]
\[-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i\]

Answer 7

now look at this (-1)^(1/4)=(-1)^((1/8)*2)
=(-1)^(2*1/8)
=1^1/8 as(-1)^2=1
=1

Answer 8

if you are doing something with laws of exponents, be advised that they work if the radicand it POSITIVE

Answer 9

again (-1)^(1/4)=(-1)^(1/2*1/2)
=(i)^1/2
=i^(1/4)

Answer 10

i have no idea what you are doing.
\[\sqrt[4]{-1}\] is not a real number

Answer 11

then tell me my mistake in answering 1. what step is wrong?

Answer 12

so for example it is not true that
\[\sqrt[4]{ab}=\sqrt[4]{a}\sqrt[4]{b}\]unless you know that a, b are positive

Answer 13

Your mistake is that you are using rules for reals numbers on an number that is not real.

Answer 14

your procedure of evaluating is very difficult

Answer 15

you are using the laws of exponents to work with radicals, but the laws of exponents do not work if the radicand is negative, that is if you are working with complex numbers. for erxample it is not true that
\[\sqrt{-5}\times \sqrt{-5} =\sqrt{-5\times -5}=\sqrt{25}=5\] this is false

Answer 16

but i think non of my step is against the rules of mathematics

Answer 17

and actually the computation for finding the solutions to \[x^4=-1\] is very simple if you look at the circle i wrote, the answers are evenly spaced around the unit circle. so you can pretty much do it in your head

Answer 18

@rosy none of your steps would be wrong if the the base was POSITIVE but it is negative, so they are all wrong

Answer 19

ok thanks for helping me now i understand.