Solve frac{-2sqrt{3}+i}{1+2sqrt{3}i}

$$\frac { - 2 \sqrt { 3 } + i } { 1 + 2 \sqrt { 3 } i }$$

Evaluate

$$i$$

Solution Steps

Rationalize the denominator of \frac{-2\sqrt{3}+i}{1+2i\sqrt{3}} by multiplying numerator and denominator by 1-2i\sqrt{3}.

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

Consider \left(1+2i\sqrt{3}\right)\left(1-2i\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}.

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

Calculate 1 to the power of 2 and get 1.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Multiply -4 and 3 to get -12.

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

Multiply -1 and -12 to get 12.

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

Add 1 and 12 to get 13.

$$\frac{\left(-2\sqrt{3}+i\right)\left(1-2i\sqrt{3}\right)}{13}$$

Apply the distributive property by multiplying each term of -2\sqrt{3}+i by each term of 1-2i\sqrt{3}.

$$\frac{-2\sqrt{3}+4i\left(\sqrt{3}\right)^{2}+i+2\sqrt{3}}{13}$$

The square of \sqrt{3} is 3.

$$\frac{-2\sqrt{3}+4i\times 3+i+2\sqrt{3}}{13}$$

Multiply 4i and 3 to get 12i.

$$\frac{-2\sqrt{3}+12i+i+2\sqrt{3}}{13}$$

Add 12i and i to get 13i.

$$\frac{-2\sqrt{3}+13i+2\sqrt{3}}{13}$$

Combine -2\sqrt{3} and 2\sqrt{3} to get 0.

$$\frac{13i}{13}$$

Divide 13i by 13 to get i.

$$i$$

Real Part

$$0$$

Solution Steps

Rationalize the denominator of \frac{-2\sqrt{3}+i}{1+2i\sqrt{3}} by multiplying numerator and denominator by 1-2i\sqrt{3}.

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

Consider \left(1+2i\sqrt{3}\right)\left(1-2i\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}.

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

Calculate 1 to the power of 2 and get 1.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Multiply -4 and 3 to get -12.

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

Multiply -1 and -12 to get 12.

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

Add 1 and 12 to get 13.

$$Re(\frac{\left(-2\sqrt{3}+i\right)\left(1-2i\sqrt{3}\right)}{13})$$

Apply the distributive property by multiplying each term of -2\sqrt{3}+i by each term of 1-2i\sqrt{3}.

$$Re(\frac{-2\sqrt{3}+4i\left(\sqrt{3}\right)^{2}+i+2\sqrt{3}}{13})$$

The square of \sqrt{3} is 3.

$$Re(\frac{-2\sqrt{3}+4i\times 3+i+2\sqrt{3}}{13})$$

Multiply 4i and 3 to get 12i.

$$Re(\frac{-2\sqrt{3}+12i+i+2\sqrt{3}}{13})$$

Add 12i and i to get 13i.

$$Re(\frac{-2\sqrt{3}+13i+2\sqrt{3}}{13})$$

Combine -2\sqrt{3} and 2\sqrt{3} to get 0.

$$Re(\frac{13i}{13})$$

Divide 13i by 13 to get i.

$$Re(i)$$

The real part of i is 0.

$$0$$