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Precalculus : Find the Roots of Complex Numbers

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Example Questions

Example Question #1 : Find The Roots Of Complex Numbers

Evaluate ${}(1+i)^{2n}$ , where is a natural number and is the complex number ${}i^{2}=-1$ .

Possible Answers:

${}2^{n+1}i^{n}$

${}-2^{n}i^{n}$

${}-2^{n+1}i^{n}$

${}2^{n}i^{n}$

${}2^{n}i^{n+1}$

Correct answer:

${}2^{n}i^{n}$

Explanation:

Note that,

${}(1+i)^{2n}=((1+i)^2)^{n}}= (2i)^{n}= 2^{n}i^{n}$

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Example Question #1 : Find The Roots Of Complex Numbers

What is the length of

$({1-i})^{3}$ ?

Possible Answers:

$2-\sqrt{2}$

$2+\sqrt{2}$

$2\sqrt{2}$

$\sqrt{2}$

Correct answer:

$2\sqrt{2}$

Explanation:

We have

$|1-i|=\sqrt{2}$ .

Hence,

$|(1-i)^3|=|1-i|^{3}=(\sqrt2)^3=2\sqrt{2}$ .

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Example Question #2 : Find The Roots Of Complex Numbers

Solve for $\small z^3=1$ (there may be more than one solution).

Possible Answers:

$\small z=1$

$\small z=1, \frac{-1+\sqrt{3}}{2}, \frac{-1-\sqrt{3}}{2}$

$\small z=-1$

$\small z=1, \frac{-1+i\sqrt{3}}{2}, \frac{-1-i\sqrt{3}}{2}$

Correct answer:

$\small z=1, \frac{-1+i\sqrt{3}}{2}, \frac{-1-i\sqrt{3}}{2}$

Explanation:

Solving that equation is equivalent to solving the roots of the polynomial $\small f(z)=z^3-1$ .

Clearly, one of roots is 1.

Thus, we can factor the polynomial as $\small f(z)=(z-1)(z^2+z+1)$

so that we solve for the roots of $\small z^2+z+1$ .

Using the quadratic equation, we solve for roots, which are $\small z=\frac{-1+i\sqrt{3}}{2}, \frac{-1-i\sqrt{3}}{2}$ .

This means the solutions to $\small z^3=1$ are

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

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Example Question #4 : Find The Roots Of Complex Numbers

Recall that $z=rcis(\theta)$ is just shorthand for $z=r(cos(\theta)+isin(\theta))$ when dealing with complex numbers in polar form.

Express $\frac{1}{(\sqrt{3}-i)^4}$ in polar form.

Possible Answers:

$2cis\left(\frac{\pi}{3}\right)$

$2cis\left(\frac{2\pi}{3}\right)$

$\frac{1}{16}cis\left(\frac{-2\pi}{3}\right)$

$\frac{1}{16}cis\left(\frac{2\pi}{3}\right)$

$2cis\left(\frac{2\pi}{12}\right)$

Correct answer:

$\frac{1}{16}cis\left(\frac{2\pi}{3}\right)$

Explanation:

First we recognize that we are trying to solve $z^{-4}$ where $z=\sqrt{3}-i$ .

Then we want to convert into polar form using,

$arg(z)=\theta =arctan(\frac{b}{a})$ and $\left | z\right |=r=\sqrt{a^2+b^2}$ .

Then since De Moivre's theorem states,

$z^n=r^ncis(n\theta)$ if is an integer, we can say

$z^{-4}=2^{-4}cis(-4*\frac{-\pi}{6})=\frac{1}{16}cis\left(\frac{2\pi}{3}\right)$ .

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Example Question #1 : Find The Roots Of Complex Numbers

Solve for z^3-z^2+z=0 (there may be more than one solution).

Possible Answers:

$0, 1, \frac{1}{2}$

$0, \frac{1}{2}\pm\frac{\sqrt{3}}{2}$

$0, \frac{1}{2}\pm\frac{\sqrt{3i}}{2}$

$0, \frac{1}{2}\pm\sqrt{3i}$

Correct answer:

$0, \frac{1}{2}\pm\frac{\sqrt{3i}}{2}$

Explanation:

To solve for the roots, just set equal to zero and solve for z using the quadratic formula ( $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ ) : $z^3-z^2+z\rightarrow z(z^2-z+1)$ and now setting both and z^2-z+1 equal to zero we end up with the answers z=0 and $z=\frac{1\pm 3i}{2}$

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Example Question #5 : Find The Roots Of Complex Numbers

Compute (3+3i)^5

Possible Answers:

972i

972

-972-972i

972-972i

972+972i

Correct answer:

-972-972i

Explanation:

To solve this question, you must first derive a few values and convert the equation into exponential form: $re^{\pi*z}$ : $r=\sqrt{9+9}=3\sqrt{2}\rightarrow tan(\theta)=\frac{3}{3}=1\rightarrow Arg(z)=\frac{\pi}{4}$

Now plug back into the original equation and solve: $(3+3i)^5=(3\sqrt{2})^5\cdot e^{\frac{i5\pi}{4}=972\sqrt{2}(cos(\frac{5\pi}{4})+isin(\frac{5\pi}{4}))=972\sqrt{2}(\frac {-\sqrt{2}{2}}-\frac ({\sqrt{2}\cdot i}{2})=-972-972i$

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Example Question #3 : Find The Roots Of Complex Numbers

Determine the length of 2i^2-1

Possible Answers:

Correct answer:

Explanation:

i^2=-1 , so $2(-1)-1=-3\rightarrow |-3|=3$

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Example Question #1 : Find The Roots Of Complex Numbers

Solve for all possible solutions to the quadratic expression: m^2-6m+12=0

Possible Answers:

$3\pm3$

$3\pm\sqrt{3}i$

$3\pm\sqrt{3}$

$3\pm3i$

Correct answer:

$3\pm\sqrt{3}i$

Explanation:

Solve for complex values of m using the aforementioned quadratic formula:

$\frac{6\pm\sqrt{36-(4)(1)(12)}}{2}$

$=\frac{6\pm\sqrt{-12}}{2}$

$=\frac{6\pm2i\sqrt{3}}{2}$

$=3\pm \sqrt{3}i$

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Example Question #1 : Find The Roots Of Complex Numbers

Which of the following lists all possible solutions to the quadratic expression: $m^{2}-6m+12=0$

Possible Answers:

$3\pm 3i$

$3\pm 3$

$3\pm \sqrt{3i}$

$3\pm \sqrt{3}$

Correct answer:

$3\pm \sqrt{3i}$

Explanation:

Solve for complex values of using the quadratic formula: $6\pm \sqrt{36-(4)(1)(12)}\rightarrow \frac{6\pm \sqrt{-12}}{2}=3\pm \sqrt{3i}$

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Example Question #10 : Find The Roots Of Complex Numbers

Determine the length of $2i^{2}-1$ .

Possible Answers:

Correct answer:

Explanation:

To begin, we must recall that $i^{2}=-1$ . Plug this in to get 2i^2-1=2(-1)-1=-2-1=-3 . Length must be a positive value, so we'll take the absolute value: $\left | -3 \right |=3$ . Therefore the length is 3.

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Precalculus : Find the Roots of Complex Numbers

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Example Questions

Example Question #1 : Find The Roots Of Complex Numbers

Example Question #1 : Find The Roots Of Complex Numbers

Example Question #2 : Find The Roots Of Complex Numbers

Example Question #4 : Find The Roots Of Complex Numbers

Express $\frac{1}{(\sqrt{3}-i)^4}$ in polar form.

Example Question #1 : Find The Roots Of Complex Numbers

Example Question #5 : Find The Roots Of Complex Numbers

Example Question #3 : Find The Roots Of Complex Numbers

Example Question #1 : Find The Roots Of Complex Numbers

Example Question #1 : Find The Roots Of Complex Numbers

Example Question #10 : Find The Roots Of Complex Numbers

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Precalculus : Find the Roots of Complex Numbers

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Example Questions

Example Question #1 : Find The Roots Of Complex Numbers

Example Question #1 : Find The Roots Of Complex Numbers

Example Question #2 : Find The Roots Of Complex Numbers

Example Question #4 : Find The Roots Of Complex Numbers

Express in polar form.

Example Question #1 : Find The Roots Of Complex Numbers

Example Question #5 : Find The Roots Of Complex Numbers

Example Question #3 : Find The Roots Of Complex Numbers

Example Question #1 : Find The Roots Of Complex Numbers

Example Question #1 : Find The Roots Of Complex Numbers

Example Question #10 : Find The Roots Of Complex Numbers

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Express $\frac{1}{(\sqrt{3}-i)^4}$ in polar form.