Begin by simplifying the equation. Divide both sides by $\displaystyle 2$ and take the square root.

$\displaystyle 2n^2=\frac{-27}{8}$

$\displaystyle n^2=\frac{-27}{16}$

$\displaystyle n=\frac{\sqrt{-27}}{\sqrt{16}}$

$\displaystyle n=\frac{3i\sqrt{3}}{4}, \frac{-3i\sqrt{3}}{4}$

Begin by simplifying the equation. Divide both sides by $\displaystyle 4$ and take the square root.

$\displaystyle 4x^2+75=0$

$\displaystyle 4x^2=-75$

$\displaystyle x^2=\frac{-75}{4}$

$\displaystyle x=\frac{\sqrt{-75}}{\sqrt{4}}$

$\displaystyle x=\frac{5i\sqrt{3}}{2}, -\frac{5i\sqrt{3}}{2}$

Solve the equation using complex numbers:

$\displaystyle 2n^2+18=0$

$\displaystyle 2n^2=-18$

$\displaystyle n^2=-9$

$\displaystyle n=\sqrt{-9}$

$\displaystyle n=3i, -3i$

Begin by simplifying the radicals using complex numbers:

$\displaystyle (\sqrt{-6})(\sqrt{-4})(\sqrt{-3})$

$\displaystyle =(i\sqrt{6})(2i)(i\sqrt{3})$ 

Multiply the factors:

$\displaystyle =(2i^2\sqrt{6})(i\sqrt{3})$

$\displaystyle =(2i^3\sqrt{18})$

Simplify. Remember that $\displaystyle i^3$ is equivalent to $\displaystyle -i$.

$\displaystyle =-2i\sqrt{18}$

$\displaystyle =-6i\sqrt{2}$

Begin by completing the square:

$\displaystyle (3i^3)(2i)^2$

$\displaystyle =(3i^3)(4i^2)$ 

Multiply the factors:

$\displaystyle =(12i^5)$ 

Simplify. Remember that $\displaystyle i^5$ is equivalent to $\displaystyle i$.

$\displaystyle =12i$

Begin by completing the square:

$\displaystyle (-2i)^4(-4i^3)$

$\displaystyle =(16i^4)(-4i^3)$ 

Now, multiply the factors:

$\displaystyle =(16i^4)(-4i^3)$

$\displaystyle =(-64i^7)$ 

Simplify. Remember that $\displaystyle i^7$ is equivalent to $\displaystyle -i$.

$\displaystyle =64i$

Use the FOIL (First, Outer, Inner, Last) to multiply the complex numbers:

$\displaystyle (7+2i)(5-3i)$

$\displaystyle =35-21i+10i-6i^2$

Combine like terms and simplify. Remember that $\displaystyle i^2$ is equivalent to $\displaystyle -1$.

$\displaystyle =35-11i-6i^2$

$\displaystyle =35-11i+6$

$\displaystyle =41-11i$

Begin by multiplying the numerator and denominator by the conjugate of the denominator:

$\displaystyle =\frac{4+3i}{1-2i}$ $\displaystyle \cdot \frac{1+2i}{1+2i}$

$\displaystyle =\frac{4+8i+3i+6i^2}{1-4i^2}$

Combine like terms:

$\displaystyle =\frac{4+11i+6i^2}{1-4i^2}$

Simplify. Remember that $\displaystyle i^2$ is equivalent to $\displaystyle -1$.

$\displaystyle =\frac{4+11i-6}{1+4}$

$\displaystyle =\frac{-2+11i}{5}$

Begin by multiplying the numerator and denominator by the conjugate of the denominator:

$\displaystyle =\frac{2-2i}{2+2i}$$\displaystyle \cdot \frac{2-2i}{2-2i}$

$\displaystyle =\frac{4-4i-4i+4i^2}{4-4i^2}$

Combine like terms:

$\displaystyle =\frac{4-8i+4i^2}{4-4i^2}$

Simplify. Remember that $\displaystyle i^2$ is equivalent to $\displaystyle -1$.

$\displaystyle =\frac{4-8i-4}{4+4}$

$\displaystyle =\frac{-8i}{8}$

$\displaystyle =-i$

Simplifying expressions with complex numbers uses exactly the same process as simplifying expressings with one variable. In this case, $\displaystyle i$ behaves similarly to any other variable. Thus, we have: 

$\displaystyle (5 + 3i) + (2 - 6i) = (5 +2) + (3 - 6)i = 7 - 3i$