Imaginary Numbers - High School Math

Card 0 of 20

Question

What is the absolute value of

Answer

The absolute value is a measure of the distance of a point from the origin. Using the pythagorean distance formula to calculate this distance.

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Question

Simplify the expression.

Answer

Combine like terms. Treat $\small i$ as if it were any other variable.

Substitute to eliminate $\small i^2$ .

Simplify.

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Question

Simplify the radical.

$\sqrt{-250}$

Answer

$\sqrt{-250}$

First, factor the term in the radical.

$\sqrt{-250}=(\sqrt{-1})(\sqrt{25})(\sqrt{10})$

Now, we can simplify.

$\sqrt{-1}=i\ \text{and}\ \sqrt{25}=5$

$(\sqrt{-1})(\sqrt{25})(\sqrt{10})=(i)(5)(\sqrt{10})$

$5i\sqrt{10}$

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Question

Multiply:

Answer

FOIL:

$= 3 \cdot 4 -3 \cdot 3i + 2i \cdot4 - 2i \cdot3i$

$= 12 -9i + 8i - 6i^{2}$

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Question

Multiply:

Answer

Since and are conmplex conjugates, they can be multiplied according to the following pattern:

$(5+4i) (5-4i) = 5 ^{2} + 4^{2} = 25 + 16 = 41$

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Question

Multiply:

$\left (7 + i \sqrt{2 } \right )\left (7 - i \sqrt{2 } \right )$

Answer

Since $7 + i \sqrt{2 }$ and $7 - i \sqrt{2 }$ are conmplex conjugates, they can be multiplied according to the following pattern:

$\left (7 + i \sqrt{2 } \right )\left (7 - i \sqrt{2 } \right ) = 7^{2} +\left ( \sqrt{2} \right )^{2} = 49 + 2 = 51$

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Question

Evaluate: $i ^{45}$

Answer

$i ^{N}$ can be evaluated by dividing by 4 and noting the remainder. Since $45 \div 4 = 11 \textrm{ R } 1$ - that is, since dividing 45 by 4 yields remainder 1:

$i ^{45} = i ^{1} = i$

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Question

Evaluate: $(4 + 3i)^{2}$

Answer

$(4 + 3i)^{2}$

$= 4^{2} + 2 \cdot 4 \cdot 3i +\left ( 3i \right )^{2}$

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Question

Which of the following is equivalent to $5i^{2}$ ?

Answer

Recall the basic property of imaginary numbers, $i^{2} = -1$ .

Keeping this in mind, $5i^{2} = 5(-1) = -5$ .

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Question

Which of the following is equivalent to:

$-4i^{4} ?$

Answer

Recall that $i^{2} = -1$ .

Then, we have that $-4(i^{4}) = -4(i^{2})^{2} = -4(-1)^{2} = -4(1) = -4$ .

Note that we used the power rule of exponents and the order of operations to simplify the exponent before multiplying by the coefficient.

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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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