Solve {left(1+i/1-iright)}^4

$${ \left( \frac{ 1+i }{ 1-i } \right) }^{ 4 }$$

Evaluate

$$1$$

Solution Steps

Multiply both numerator and denominator of \frac{1+i}{1-i} by the complex conjugate of the denominator, 1+i.

$$\left(\frac{\left(1+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}\right)^{4}$$

Do the multiplications in \frac{\left(1+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}.

$$\left(\frac{2i}{2}\right)^{4}$$

Divide 2i by 2 to get i.

$$i^{4}$$

Calculate i to the power of 4 and get 1.

$$1$$

Real Part

$$1$$

Solution Steps

Multiply both numerator and denominator of \frac{1+i}{1-i} by the complex conjugate of the denominator, 1+i.

$$Re(\left(\frac{\left(1+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}\right)^{4})$$

Do the multiplications in \frac{\left(1+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}.

$$Re(\left(\frac{2i}{2}\right)^{4})$$

Divide 2i by 2 to get i.

$$Re(i^{4})$$

Calculate i to the power of 4 and get 1.

$$Re(1)$$

The real part of 1 is 1.

$$1$$