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\frac{x\left(1-i\sqrt{3}\right)}{\left(1+i\sqrt{3}\right)\left(1-i\sqrt{3}\right)}=2+2i
Rationalize the denominator of \frac{x}{1+i\sqrt{3}} by multiplying numerator and denominator by 1-i\sqrt{3}.
\frac{x\left(1-i\sqrt{3}\right)}{1^{2}-\left(i\sqrt{3}\right)^{2}}=2+2i
Consider \left(1+i\sqrt{3}\right)\left(1-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}.
\frac{x\left(1-i\sqrt{3}\right)}{1-\left(i\sqrt{3}\right)^{2}}=2+2i
Calculate 1 to the power of 2 and get 1.
\frac{x\left(1-i\sqrt{3}\right)}{1-i^{2}\left(\sqrt{3}\right)^{2}}=2+2i
Expand \left(i\sqrt{3}\right)^{2}.
\frac{x\left(1-i\sqrt{3}\right)}{1-\left(-\left(\sqrt{3}\right)^{2}\right)}=2+2i
Calculate i to the power of 2 and get -1.
\frac{x\left(1-i\sqrt{3}\right)}{1-\left(-3\right)}=2+2i
The square of \sqrt{3} is 3.
\frac{x\left(1-i\sqrt{3}\right)}{1+3}=2+2i
Multiply -1 and -3 to get 3.
\frac{x\left(1-i\sqrt{3}\right)}{4}=2+2i
Add 1 and 3 to get 4.
\frac{x-ix\sqrt{3}}{4}=2+2i
Use the distributive property to multiply x by 1-i\sqrt{3}.
x-ix\sqrt{3}=\left(2+2i\right)\times 4
Multiply both sides by 4.
-\sqrt{3}ix+x=4\left(2+2i\right)
Reorder the terms.
-i\sqrt{3}x+x=4\left(2+2i\right)
Multiply -1 and i to get -i.
-i\sqrt{3}x+x=4\times 2+4\times \left(2i\right)
Multiply 4 times 2+2i.
-i\sqrt{3}x+x=8+8i
Do the multiplications in 4\times 2+4\times \left(2i\right).
\left(-i\sqrt{3}+1\right)x=8+8i
Combine all terms containing x.
\left(1-\sqrt{3}i\right)x=8+8i
Reorder the terms.
\left(-\sqrt{3}i+1\right)x=8+8i
The equation is in standard form.
\frac{\left(-\sqrt{3}i+1\right)x}{-\sqrt{3}i+1}=\frac{8+8i}{-\sqrt{3}i+1}
Divide both sides by 1-i\sqrt{3}.
x=\frac{8+8i}{-\sqrt{3}i+1}
Dividing by 1-i\sqrt{3} undoes the multiplication by 1-i\sqrt{3}.
x=\sqrt{3}\left(-2+2i\right)+\left(2+2i\right)
Divide 8+8i by 1-i\sqrt{3}.