Question

x4 - 1/x4=0..solve the following ​

Answer 1

Answer:

Using this, x4=−1=eiπ=e(2n+1)πi as e2mπi=1 where m,n are integers.

So, x=e(2n+1)πi4=cos(2n+1)π4+isin(2n+1)π4 where n has any 4 in-congruent values (mod4), the most simple set of values of n can be {0,1,2,3}.

If xm=e(2m+1)πi4,xm+2=e(2m+3)πi4=e(2m+1)πi4⋅eiπ2=−xm

Also, observe that if y is a solution of x4=−1, so is −y

x0=cosπ4+isinπ4=1+i2√

x1=cos3π4+isin3π4=−1+i2√

x2=−x0

x3=−x1

So, the values of x are ±(1+i2√),±(−1+i2√)

Step-by-step explanation:

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Answer 2

Answer:

Using this, x4=−1=eiπ=e(2n+1)πi as e2mπi=1 where m,n are integers.

So, x=e(2n+1)πi4=cos(2n+1)π4+isin(2n+1)π4 where n has any 4 in-congruent values (mod4), the most simple set of values of n can be {0,1,2,3}.

If xm=e(2m+1)πi4,xm+2=e(2m+3)πi4=e(2m+1)πi4⋅eiπ2=−xm

Also, observe that if y is a solution of x4=−1, so is −y

x0=cosπ4+isinπ4=1+i2√

x1=cos3π4+isin3π4=−1+i2√

x2=−x0

x3=−x1

So, the values of x are ±(1+i2√),±(−1+i2√)