Solve (10+isqrt{2})*(10-isqrt{2})

$$( 10 + i \sqrt { 2 } ) \cdot ( 10 - i \sqrt { 2 } )$$

Evaluate

$$102$$

Solution Steps

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

$$10^{2}-\left(i\sqrt{2}\right)^{2}$$

Calculate 10 to the power of 2 and get 100.

$$100-\left(i\sqrt{2}\right)^{2}$$

Expand \left(i\sqrt{2}\right)^{2}.

$$100-i^{2}\left(\sqrt{2}\right)^{2}$$

Calculate i to the power of 2 and get -1.

$$100-\left(-\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$100-\left(-2\right)$$

The opposite of -2 is 2.

$$100+2$$

Add 100 and 2 to get 102.

$$102$$

Real Part

$$102$$

Solution Steps

Consider \left(10+i\sqrt{2}\right)\left(10-i\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$Re(10^{2}-\left(i\sqrt{2}\right)^{2})$$

Calculate 10 to the power of 2 and get 100.

$$Re(100-\left(i\sqrt{2}\right)^{2})$$

Expand \left(i\sqrt{2}\right)^{2}.

$$Re(100-i^{2}\left(\sqrt{2}\right)^{2})$$

Calculate i to the power of 2 and get -1.

$$Re(100-\left(-\left(\sqrt{2}\right)^{2}\right))$$

The square of \sqrt{2} is 2.

$$Re(100-\left(-2\right))$$

The opposite of -2 is 2.

$$Re(100+2)$$

Add 100 and 2 to get 102.

$$Re(102)$$

The real part of 102 is 102.

$$102$$