Solve =(1+1)(1-i)/(1-i)(1-i)

$$= \frac { ( 1 + 1 ) ( 1 - i ) } { ( 1 - i ) ( 1 - i ) }$$

Evaluate

$$1+i$$

Solution Steps

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{1+1}{\left(1-i\right)\left(1-i\right)^{0}}$$

Add 1 and 1 to get 2.

$$\frac{2}{\left(1-i\right)\left(1-i\right)^{0}}$$

To multiply powers of the same base, add their exponents. Add 1 and 0 to get 1.

$$\frac{2}{\left(1-i\right)^{1}}$$

Calculate 1-i to the power of 1 and get 1-i.

$$\frac{2}{1-i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 1+i.

$$\frac{2\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\left(1+i\right)}{1^{2}-i^{2}}$$

By definition, i^{2} is -1. Calculate the denominator.

$$\frac{2\left(1+i\right)}{2}$$

Multiply 2 times 1+i.

$$\frac{2\times 1+2i}{2}$$

Do the multiplications in 2\times 1+2i.

$$\frac{2+2i}{2}$$

Divide 2+2i by 2 to get 1+i.

$$1+i$$

Real Part

$$1$$

Solution Steps

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$Re(\frac{1+1}{\left(1-i\right)\left(1-i\right)^{0}})$$

Add 1 and 1 to get 2.

$$Re(\frac{2}{\left(1-i\right)\left(1-i\right)^{0}})$$

To multiply powers of the same base, add their exponents. Add 1 and 0 to get 1.

$$Re(\frac{2}{\left(1-i\right)^{1}})$$

Calculate 1-i to the power of 1 and get 1-i.

$$Re(\frac{2}{1-i})$$

Multiply both numerator and denominator of \frac{2}{1-i} by the complex conjugate of the denominator, 1+i.

$$Re(\frac{2\left(1+i\right)}{\left(1-i\right)\left(1+i\right)})$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$Re(\frac{2\left(1+i\right)}{1^{2}-i^{2}})$$

By definition, i^{2} is -1. Calculate the denominator.

$$Re(\frac{2\left(1+i\right)}{2})$$

Multiply 2 times 1+i.

$$Re(\frac{2\times 1+2i}{2})$$

Do the multiplications in 2\times 1+2i.

$$Re(\frac{2+2i}{2})$$

Divide 2+2i by 2 to get 1+i.

$$Re(1+i)$$

The real part of 1+i is 1.

$$1$$