Solve =sqrt{2}(-1/sqrt{2}+1/sqrt{2}i)text{or}sqrt{2}(-1/sqrt{2}-1/sqrt{2}i)

$$= \sqrt { 2 } ( - \frac { 1 } { \sqrt { 2 } } + \frac { 1 } { \sqrt { 2 } } i ) \text { or } \sqrt { 2 } ( - \frac { 1 } { \sqrt { 2 } } - \frac { 1 } { \sqrt { 2 } } i )$$

Evaluate

$$-1+i$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Use the distributive property to multiply \sqrt{2} by -\frac{\sqrt{2}}{2}+i\times \frac{\sqrt{2}}{2}.

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

Express \sqrt{2}\left(-\frac{\sqrt{2}}{2}\right) as a single fraction.

$$\frac{-\sqrt{2}\sqrt{2}}{2}+i\sqrt{2}\times \frac{\sqrt{2}}{2}$$

Express \sqrt{2}\times \frac{\sqrt{2}}{2} as a single fraction.

$$\frac{-\sqrt{2}\sqrt{2}}{2}+i\times \frac{\sqrt{2}\sqrt{2}}{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{-2}{2}+i\times \frac{\sqrt{2}\sqrt{2}}{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Divide 2 by 2 to get 1.

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

Cancel out 2 and 2.

$$-1+i$$

Real Part

$$-1$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Use the distributive property to multiply \sqrt{2} by -\frac{\sqrt{2}}{2}+i\times \frac{\sqrt{2}}{2}.

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

Express \sqrt{2}\left(-\frac{\sqrt{2}}{2}\right) as a single fraction.

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

Express \sqrt{2}\times \frac{\sqrt{2}}{2} as a single fraction.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Divide 2 by 2 to get 1.

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

Cancel out 2 and 2.

$$Re(-1+i)$$

The real part of -1+i is -1.

$$-1$$