Solve 8/2-2sqrt{3i}*frac{2-2sqrt{3}i}{2-2sqrt{3}i}

$$\frac { 8 } { 2 - 2 \sqrt { 3 } i } \cdot \frac { 2 - 2 \sqrt { 3 } i } { 2 - 2 \sqrt { 3 } i }$$

Evaluate

$$1+\sqrt{3}i\approx 1+1.732050808i$$

Solution Steps

Divide 2-2i\sqrt{3} by 2-2i\sqrt{3} to get 1.

$$\frac{8}{2-2i\sqrt{3}}\times 1$$

Rationalize the denominator of \frac{8}{2-2i\sqrt{3}} by multiplying numerator and denominator by 2+2i\sqrt{3}.

$$\frac{8\left(2+2i\sqrt{3}\right)}{\left(2-2i\sqrt{3}\right)\left(2+2i\sqrt{3}\right)}\times 1$$

Consider \left(2-2i\sqrt{3}\right)\left(2+2i\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}.

$$\frac{8\left(2+2i\sqrt{3}\right)}{2^{2}-\left(-2i\sqrt{3}\right)^{2}}\times 1$$

Calculate 2 to the power of 2 and get 4.

$$\frac{8\left(2+2i\sqrt{3}\right)}{4-\left(-2i\sqrt{3}\right)^{2}}\times 1$$

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

$$\frac{8\left(2+2i\sqrt{3}\right)}{4-\left(-2i\right)^{2}\left(\sqrt{3}\right)^{2}}\times 1$$

Calculate -2i to the power of 2 and get -4.

$$\frac{8\left(2+2i\sqrt{3}\right)}{4-\left(-4\left(\sqrt{3}\right)^{2}\right)}\times 1$$

The square of \sqrt{3} is 3.

$$\frac{8\left(2+2i\sqrt{3}\right)}{4-\left(-4\times 3\right)}\times 1$$

Multiply -4 and 3 to get -12.

$$\frac{8\left(2+2i\sqrt{3}\right)}{4-\left(-12\right)}\times 1$$

Multiply -1 and -12 to get 12.

$$\frac{8\left(2+2i\sqrt{3}\right)}{4+12}\times 1$$

Add 4 and 12 to get 16.

$$\frac{8\left(2+2i\sqrt{3}\right)}{16}\times 1$$

Divide 8\left(2+2i\sqrt{3}\right) by 16 to get \frac{1}{2}\left(2+2i\sqrt{3}\right).

$$\frac{1}{2}\left(2+2i\sqrt{3}\right)\times 1$$

Use the distributive property to multiply \frac{1}{2} by 2+2i\sqrt{3}.

$$\left(\frac{1}{2}\times 2+\frac{1}{2}\times \left(2i\right)\sqrt{3}\right)\times 1$$

Cancel out 2 and 2.

$$\left(1+\frac{1}{2}\times \left(2i\right)\sqrt{3}\right)\times 1$$

Multiply \frac{1}{2} and 2i to get i.

$$\left(1+i\sqrt{3}\right)\times 1$$

Use the distributive property to multiply 1+i\sqrt{3} by 1.

$$1+i\sqrt{3}$$

Factor

$$1+\sqrt{3}i$$