Question

Write in standard notation (1/sqrt2-1/sqrt2 i)^6

Answer 1

The expression (1/sqrt(2) - 1/sqrt(2) i)^6 simplifies to the complex number -i in standard notation.

Step-by-step explanation:

To write the expression (1/sqrt(2) - 1/sqrt(2) i) to the sixth power in standard notation, we first need to recognize that the expression is complex and in the form of a - bi, where a and b are both 1/sqrt(2). Since the absolute value (modulus) of this complex number is 1 (because sqrt(a^2 + b^2) = sqrt((1/sqrt(2))^2 + (1/sqrt(2))^2) = sqrt(1/2 + 1/2) = sqrt(1) = 1), raising it to the sixth power means effectively rotating it on the complex plane. The angle that the complex number makes with the positive real axis is 45 degrees (or pi/4 radians), and when we raise it to the sixth power, we multiply this angle by 6. Therefore, the argument of the complex number after raising it to the power of 6 is 6*(pi/4), which simplifies to 3pi/2. This results in a complex number on the negative imaginary axis. Since the modulus is 1, the standard notation of this complex number is 0 - i, which can simply be written as -i.

Answer 2

Hello,

z=(1/√2-1/√2 i)^6=1/8*(1-i)^6=1/8*(√2∡-45°)^6=1/8*8∡-270°=1∡90°=i
======================================================