Solve (2+sqrt{3}i)(2-sqrt{3}i)

$$( 2 + \sqrt { 3 } i ) ( 2 - \sqrt { 3 } i )$$

Evaluate

$$7$$

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}.

$$2^{2}-\left(i\sqrt{3}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4-\left(i\sqrt{3}\right)^{2}$$

Expand \left(i\sqrt{3}\right)^{2}.

$$4-i^{2}\left(\sqrt{3}\right)^{2}$$

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

$$4-\left(-\left(\sqrt{3}\right)^{2}\right)$$

The square of \sqrt{3} is 3.

$$4-\left(-3\right)$$

The opposite of -3 is 3.

$$4+3$$

Add 4 and 3 to get 7.

$$7$$

Real Part

$$7$$

Solution Steps

Consider \left(2+i\sqrt{3}\right)\left(2-i\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$Re(2^{2}-\left(i\sqrt{3}\right)^{2})$$

Calculate 2 to the power of 2 and get 4.

$$Re(4-\left(i\sqrt{3}\right)^{2})$$

Expand \left(i\sqrt{3}\right)^{2}.

$$Re(4-i^{2}\left(\sqrt{3}\right)^{2})$$

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

$$Re(4-\left(-\left(\sqrt{3}\right)^{2}\right))$$

The square of \sqrt{3} is 3.

$$Re(4-\left(-3\right))$$

The opposite of -3 is 3.

$$Re(4+3)$$

Add 4 and 3 to get 7.

$$Re(7)$$

The real part of 7 is 7.

$$7$$