Using Expressions with Complex Numbers - High School Math

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Question

Simplify $\frac{4}{3+2i}$

Answer

Multiplying top and bottom by the complex conjugate eliminates i from the denominator

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Question

Multiply, then simplify.

Answer

Use FOIL to multiply. We can treat $\small i$ as a variable for now, just as if it were an $\small x$ .

Combine like terms.

Now we can substitute for $\small i^2$ .

Simplify.

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Question

Which of the following is equivalent to $\frac{3+i}{3-4i}$ , where $i=\sqrt{-1}$ ?

Answer

We will need to simplify the rational expression by removing imaginary terms from the denominator. Often in such problems, we want to multiply the numerator and denominator by the conjugate of the denominator, which will usually eliminate the imaginary term from the denominator.

In this problem, the denominator is . Remember that, in general, the conjugate of the complex number is equal to , where a and b are both nonzero constants. Thus, the conjugate of is equal to .

We need to multiply both the numerator and denominator of the fraction $\frac{3+i}{3-4i}$ by .

$\frac{3+i}{3-4i}=\frac{3+i}{3-4i}\cdot\frac{3+4i}{3+4i}=\frac{(3+i)(3+4i)}{(3-4i)(3+4i)}$

Next, we use the FOIL method to simplify both the numerator and denominator. The FOIL method of binomial multiplication requires us to mutiply together the first, outside, inner, and last terms of each binomial and then sum them.

$\frac{(3+i)(3+4i)}{(3-4i)(3+4i)}=\frac{3(3)+3(4i)+i(3)+i(4i)}{3(3)+3(4i)-4i(3)+(-4i)(4i)}$

Now, we can start simplifying.

$\frac{3(3)+3(4i)+i(3)+i(4i)}{3(3)+3(4i)-4i(3)+(-4i)(4i)}=\frac{9+12i+3i+4i^2}{9+12i-12i-16i^2}$

Use the fact that $i^2=(\sqrt{-1})^2=-1$ .

$\frac{9+12i+3i+4i^2}{9+12i-12i-16i^2}=\frac{9+12i+3i+4(-1)}{9+12i-12i-16(-1)}$

$=\frac{9+15i-4}{9+16}=\frac{5+15i}{25}=\frac{5}{25}+\frac{15i}{25}$

$=\frac{1}{5}+\frac{3}{5}i$

The answer is $\frac{1}{5}+\frac{3}{5}i$ .

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Question

Simplify the following complex number expression:

$(\sqrt{-8})(\sqrt{-6})$

Answer

Begin by factoring the radical expression using imaginary numbers:

$(\sqrt{-8})(\sqrt{-6})$

$(2i\sqrt{2})(i\sqrt{6})$

Now, multiply the factors:

$(2i\sqrt{2})(i\sqrt{6}) = 2i^2\sqrt{12}$

Simplify. Remember that is equivalent to

$4i^2\sqrt{3}$

$-4\sqrt{3}$

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Question

Simplify the following complex number expression:

Answer

Begin by completing the square:

Now, multiply the factors. Remember that is equivalent to

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Question

Solvethe following complex number expression for :

Answer

Solve the equation using complex numbers:

$n=\sqrt{-9}$

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Question

Simplify the following complex number expression:

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

Answer

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

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

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

Combine like terms:

$\frac{3+9i+6i^2}{1-4i^2}$

Simplify. Remember that is equivalent to

$\frac{3+9i-6}{1+4}$

$\frac{-3+9i}{5}$

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Question

Simplify the following complex number expression:

$(\sqrt{-8})(\sqrt{-12})$

Answer

Begin by factoring the radical expression using imaginary numbers:

$(\sqrt{-8})(\sqrt{-12})$

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

Now, multiply the factors:

$4i^2\sqrt{6}$

Simplify. Remember that is equivalent to

$-4\sqrt{6}$

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Question

Simplify the following complex number expression:

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

Answer

Begin by simplifying the radicals using complex numbers:

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

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

Multiply the factors:

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

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

Simplify. Remember that is equivalent to .

$=-2i\sqrt{18}$

$=-6i\sqrt{2}$

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Question

Simplify the following complex number expression:

Answer

Begin by completing the square:

Multiply the factors:

Simplify. Remember that is equivalent to .

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Question

Simplify the following complex number expression:

Answer

Begin by completing the square:

Now, multiply the factors:

Simplify. Remember that is equivalent to .

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Question

Simplify the following complex number expression:

Answer

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

Combine like terms and simplify. Remember that is equivalent to .

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Question

Simplify the following complex number expression:

Answer

Use the FOIL (First, Outer, Inner, Last) method to expand the expression:

Simplify. Remember that is equivalent to .

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Question

Simplify the following complex number expression:

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

Answer

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

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

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

Combine like terms:

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

Simplify. Remember that is equivalent to .

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

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

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Question

Simplify the following complex number expression:

$\frac{2-2i}{2+2i}$

Answer

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

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

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

Combine like terms:

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

Simplify. Remember that is equivalent to .

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

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

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Question

Solve the following complex number equation:

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

Answer

Begin by simplifying the equation. Divide both sides by and take the square root.

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

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

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

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

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Question

Solve the following complex number equation:

Answer

Begin by simplifying the equation. Divide both sides by and take the square root.

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

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

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

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Question

Simplify the following expression:

Answer

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

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