[latex]\begin{align}&r=\sqrt{{x}^{2}+{y}^{2}} \\ &r=\sqrt{{0}^{2}+{4}^{2}} \\ &r=\sqrt{16} \\ &r=4 \end{align}[/latex]. This in general is written for any complex number as: The product of two complex numbers in polar form is found by _____ their moduli and _____ their arguments multiplying, adding r₁(cosθ₁+i sinθ₁)/r₂(cosθ₂+i sinθ₂)= The polar form or trigonometric form of a complex number P is z = r (cos θ + i sin θ) The value "r" represents the absolute value or modulus of the complex number … The absolute value [latex]z[/latex] is 5. The rectangular form of the given number in complex form is [latex]12+5i[/latex]. Many amazing properties of complex numbers are revealed by looking at them in polar form!Let’s learn how to convert a complex number into polar … If [latex]{z}_{1}={r}_{1}\left(\cos {\theta }_{1}+i\sin {\theta }_{1}\right)[/latex] and [latex]{z}_{2}={r}_{2}\left(\cos {\theta }_{2}+i\sin {\theta }_{2}\right)[/latex], then the product of these numbers is given as: [latex]\begin{align}{z}_{1}{z}_{2}&={r}_{1}{r}_{2}\left[\cos \left({\theta }_{1}+{\theta }_{2}\right)+i\sin \left({\theta }_{1}+{\theta }_{2}\right)\right] \\ {z}_{1}{z}_{2}&={r}_{1}{r}_{2}\text{cis}\left({\theta }_{1}+{\theta }_{2}\right) \end{align}[/latex]. For the rest of this section, we will work with formulas developed by French mathematician Abraham de Moivre (1667-1754). The n th Root Theorem Complex Numbers In Polar Form De Moivre's Theorem, Products, Quotients, Powers, and nth Roots Prec - Duration: 1:14:05. To better understand the product of complex numbers, we first investigate the trigonometric (or polar) form of a complex number. Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. Converting between the algebraic form ( + ) and the polar form of complex numbers is extremely useful. Plot the complex number [latex]2 - 3i[/latex] in the complex plane. Plot the point in the complex plane by moving [latex]a[/latex] units in the horizontal direction and [latex]b[/latex] units in the vertical direction. Substitute the results into the formula: [latex]z=r\left(\cos \theta +i\sin \theta \right)[/latex]. θ is the argument of the complex number. Then, multiply through by [latex]r[/latex]. Where: 2. Using the formula [latex]\tan \theta =\frac{y}{x}[/latex] gives, [latex]\begin{align}&\tan \theta =\frac{1}{1} \\ &\tan \theta =1 \\ &\theta =\frac{\pi }{4} \end{align}[/latex]. Find the rectangular form of the complex number given [latex]r=13[/latex] and [latex]\tan \theta =\frac{5}{12}[/latex]. Subtraction is... To multiply complex numbers in polar form, multiply the magnitudes and add the angles. This polar form is represented with the help of polar coordinates of real and imaginary numbers in the coordinate system. \\ &{z}^{\frac{1}{3}}=2\left(\cos \left(\frac{8\pi }{9}\right)+i\sin \left(\frac{8\pi }{9}\right)\right) \end{align}[/latex], [latex]\begin{align}&{z}^{\frac{1}{3}}=2\left[\cos \left(\frac{2\pi }{9}+\frac{12\pi }{9}\right)+i\sin \left(\frac{2\pi }{9}+\frac{12\pi }{9}\right)\right]&& \text{Add }\frac{2\left(2\right)\pi }{3}\text{ to each angle.} Find θ1 − θ2. Hence, the polar form of 7-5i is represented by: Suppose we have two complex numbers, one in a rectangular form and one in polar form. Complex Number Calculator The calculator will simplify any complex expression, with steps shown. [latex]z=3\left(\cos \left(\frac{\pi }{2}\right)+i\sin \left(\frac{\pi }{2}\right)\right)[/latex]. We begin by evaluating the trigonometric expressions. The only qualification is that all variables must be expressed in complex form, taking into account phase as well as magnitude, and all voltages and currents must be of the same frequency (in order that their phas… The absolute value of a complex number is the same as its magnitude. In this explainer, we will discover how converting to polar form can seriously simplify certain calculations with complex numbers. Notice that the absolute value of a real number gives the distance of the number from 0, while the absolute value of a complex number gives the distance of the number from the origin, [latex]\left(0,\text{ }0\right)[/latex]. Finding Roots of Complex Numbers in Polar Form. Plotting a complex number [latex]a+bi[/latex] is similar to plotting a real number, except that the horizontal axis represents the real part of the number, [latex]a[/latex], and the vertical axis represents the imaginary part of the number, [latex]bi[/latex]. [latex]\begin{gathered}\cos \left(\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}\\\sin \left(\frac{\pi }{6}\right)=\frac{1}{2}\end{gathered}[/latex], After substitution, the complex number is, [latex]z=12\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)[/latex], [latex]\begin{align}z&=12\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i\right) \\ &=\left(12\right)\frac{\sqrt{3}}{2}+\left(12\right)\frac{1}{2}i \\ &=6\sqrt{3}+6i \end{align}[/latex]. Write [latex]z=\sqrt{3}+i[/latex] in polar form. Given [latex]z=x+yi[/latex], a complex number, the absolute value of [latex]z[/latex] is defined as, [latex]|z|=\sqrt{{x}^{2}+{y}^{2}}[/latex]. Substituting, we have. In the last tutorial about Phasors, we saw that a complex number is represented by a real part and an imaginary part that takes the generalised form of: 1. To find the quotient of two complex numbers in polar form, find the quotient of the two moduli and the difference of the two angles. To convert into polar form modulus and argument of the given complex number, i.e. We add [latex]\frac{2k\pi }{n}[/latex] to [latex]\frac{\theta }{n}[/latex] in order to obtain the periodic roots. Get the free "Convert Complex Numbers to Polar Form" widget for your website, blog, Wordpress, Blogger, or iGoogle. Because and because lies in Quadrant III, you choose θ to be θ = π + π/3 = 4π/3. There are several ways to represent a formula for finding roots of complex numbers in polar form. When we use these formulas, we turn a complex number, a + bi, into its polar form of z = r (cos (theta) + i*sin (theta)) where a = r*cos (theta) and b = r*sin (theta). Find the polar form of [latex]-4+4i[/latex]. Convert the polar form of the given complex number to rectangular form: [latex]z=12\left(\cos \left(\frac{\pi }{6}\right)+i\sin \left(\frac{\pi }{6}\right)\right)[/latex]. Your email address will not be published. Enter ( 6 + 5 . ) Next, we look at [latex]x[/latex]. [latex]\begin{align}z&=13\left(\cos \theta +i\sin \theta \right) \\ &=13\left(\frac{12}{13}+\frac{5}{13}i\right) \\ &=12+5i \end{align}[/latex]. Given [latex]z=1 - 7i[/latex], find [latex]|z|[/latex]. The Organic Chemistry Tutor 364,283 views If [latex]z=r\left(\cos \theta +i\sin \theta \right)[/latex] is a complex number, then, [latex]\begin{align}&{z}^{n}={r}^{n}\left[\cos \left(n\theta \right)+i\sin \left(n\theta \right)\right]\\ &{z}^{n}={r}^{n}\text{cis}\left(n\theta \right)\end{align}[/latex]. Lets connect three AC voltage sources in series and use complex numbers to determine additive voltages. Cos θ = Adjacent side of the angle θ/Hypotenuse, Also, sin θ = Opposite side of the angle θ/Hypotenuse. Multiplication of complex numbers is more complicated than addition of complex numbers. Find quotients of complex numbers in polar form. REVIEW: To add complex numbers in rectangular form, add the real components and add the imaginary components. Let us learn here, in this article, how to derive the polar form of complex numbers. Find the absolute value of the complex number [latex]z=12 - 5i[/latex]. If then becomes $e^ {i\theta}=\cos {\theta}+i\sin {\theta} Do … Usually, we represent the complex numbers, in the form of z = x+iy where ‘i’ the imaginary number. When [latex]k=0[/latex], we have, [latex]{z}^{\frac{1}{3}}=2\left(\cos \left(\frac{2\pi }{9}\right)+i\sin \left(\frac{2\pi }{9}\right)\right)[/latex], [latex]\begin{align}&{z}^{\frac{1}{3}}=2\left[\cos \left(\frac{2\pi }{9}+\frac{6\pi }{9}\right)+i\sin \left(\frac{2\pi }{9}+\frac{6\pi }{9}\right)\right] && \text{ Add }\frac{2\left(1\right)\pi }{3}\text{ to each angle.} Given a complex number in rectangular form expressed as [latex]z=x+yi[/latex], we use the same conversion formulas as we do to write the number in trigonometric form: We review these relationships in Figure 5. Replace r with r1 r2, and replace θ with θ1 − θ2. The polar form of a complex number is another way to represent a complex number. The rectangular form of the given point in complex form is [latex]6\sqrt{3}+6i[/latex]. Evaluate the trigonometric functions, and multiply using the distributive property. To divide complex numbers in polar form we need to divide the moduli and subtract the arguments. The polar form of a complex number is another way of representing complex numbers.. Use De Moivre’s Theorem to evaluate the expression. Each complex number corresponds to a point (a, b) in the complex plane. Finding Roots of Complex Numbers in Polar Form To find the nth root of a complex number in polar form, we use the Root Theorem or De Moivre’s Theorem and raise the complex number to a power with a rational exponent. We first encountered complex numbers in Precalculus I. Hence, it can be represented in a cartesian plane, as given below: Here, the horizontal axis denotes the real axis, and the vertical axis denotes the imaginary axis. Plot the point [latex]1+5i[/latex] in the complex plane. The polar form of a complex number is. Find the product and the quotient of [latex]{z}_{1}=2\sqrt{3}\left(\cos \left(150^\circ \right)+i\sin \left(150^\circ \right)\right)[/latex] and [latex]{z}_{2}=2\left(\cos \left(30^\circ \right)+i\sin \left(30^\circ \right)\right)[/latex]. NOTE: If you set the calculator to return polar form, you can press Enter and the calculator will convert this number to polar form. To find the nth root of a complex number in polar form, we use the [latex]n\text{th}[/latex] Root Theorem or De Moivre’s Theorem and raise the complex number to a power with a rational exponent. A complex number on the polar form can be expressed as Z = r (cosθ + j sinθ) (3) where r = modulus (or magnitude) of Z - and is written as "mod Z" or |Z| θ = argument(or amplitude) of Z - and is written as "arg Z" r can be determined using Pythagoras' theorem r = (a2 + b2)1/2(4) θcan be determined by trigonometry θ = tan-1(b / a) (5) (3)can also be expressed as Z = r ej θ(6) As we can se from (1), (3) and (6) - a complex number can be written in three different ways. Evaluate the expression [latex]{\left(1+i\right)}^{5}[/latex] using De Moivre’s Theorem. The absolute value of a complex number is the same as its magnitude, or [latex]|z|[/latex]. [latex]\begin{align}&\frac{{z}_{1}}{{z}_{2}}=\frac{2}{4}\left[\cos \left(213^\circ -33^\circ \right)+i\sin \left(213^\circ -33^\circ \right)\right] \\ &\frac{{z}_{1}}{{z}_{2}}=\frac{1}{2}\left[\cos \left(180^\circ \right)+i\sin \left(180^\circ \right)\right] \\ &\frac{{z}_{1}}{{z}_{2}}=\frac{1}{2}\left[-1+0i\right] \\ &\frac{{z}_{1}}{{z}_{2}}=-\frac{1}{2}+0i \\ &\frac{{z}_{1}}{{z}_{2}}=-\frac{1}{2} \end{align}[/latex]. The real and complex components of coordinates are found in terms of r and θ where r is the length of the vector, and θ is the angle made with the real axis. Then, multiply through by [latex]r[/latex]. When dividing complex numbers in polar form, we divide the r terms and subtract the angles. But in polar form, the complex numbers are represented as the combination of modulus and argument. Nonzero complex numbers written in polar form are equal if and only if they have the same magnitude and their arguments differ by an integer multiple of 2 π . Next, we will look at how we can describe a complex number slightly differently – instead of giving the and coordinates, we will give a distance (the modulus) and angle (the argument). Every real number graphs to a unique point on the real axis. Find the quotient of [latex]{z}_{1}=2\left(\cos \left(213^\circ \right)+i\sin \left(213^\circ \right)\right)[/latex] and [latex]{z}_{2}=4\left(\cos \left(33^\circ \right)+i\sin \left(33^\circ \right)\right)[/latex]. Thus, the polar form is Let us consider (x, y) are the coordinates of complex numbers x+iy. Entering complex numbers in rectangular form: To enter: 6+5j in rectangular form. It is the distance from the origin to the point: [latex]|z|=\sqrt{{a}^{2}+{b}^{2}}[/latex]. To convert from polar form to rectangular form, first evaluate the trigonometric functions. It is also in polar form. In this section, we will focus on the mechanics of working with complex numbers: translation of complex numbers from polar form to rectangular form and vice versa, interpretation of complex numbers in the scheme of applications, and application of De Moivre’s Theorem. Find the product of [latex]{z}_{1}{z}_{2}[/latex], given [latex]{z}_{1}=4\left(\cos \left(80^\circ \right)+i\sin \left(80^\circ \right)\right)[/latex] and [latex]{z}_{2}=2\left(\cos \left(145^\circ \right)+i\sin \left(145^\circ \right)\right)[/latex]. r and θ. Converting Complex Numbers to Polar Form. \displaystyle z= r (\cos {\theta}+i\sin {\theta)} . Z - is the Complex Number representing the Vector 3. x - is the Real part or the Active component 4. y - is the Imaginary part or the Reactive component 5. j - is defined by √-1In the rectangular form, a complex number can be represented as a point on a two dimensional plane calle… If [latex]{z}_{1}={r}_{1}\left(\cos {\theta }_{1}+i\sin {\theta }_{1}\right)[/latex] and [latex]{z}_{2}={r}_{2}\left(\cos {\theta }_{2}+i\sin {\theta }_{2}\right)[/latex], then the quotient of these numbers is, [latex]\begin{align}&\frac{{z}_{1}}{{z}_{2}}=\frac{{r}_{1}}{{r}_{2}}\left[\cos \left({\theta }_{1}-{\theta }_{2}\right)+i\sin \left({\theta }_{1}-{\theta }_{2}\right)\right],{z}_{2}\ne 0\\ &\frac{{z}_{1}}{{z}_{2}}=\frac{{r}_{1}}{{r}_{2}}\text{cis}\left({\theta }_{1}-{\theta }_{2}\right),{z}_{2}\ne 0\end{align}[/latex]. Notice that the moduli are divided, and the angles are subtracted. The modulus, then, is the same as [latex]r[/latex], the radius in polar form. The rules are based on multiplying the moduli and adding the arguments. [latex]\begin{align}&{z}_{1}{z}_{2}=4\cdot 2\left[\cos \left(80^\circ +145^\circ \right)+i\sin \left(80^\circ +145^\circ \right)\right] \\ &{z}_{1}{z}_{2}=8\left[\cos \left(225^\circ \right)+i\sin \left(225^\circ \right)\right] \\ &{z}_{1}{z}_{2}=8\left[\cos \left(\frac{5\pi }{4}\right)+i\sin \left(\frac{5\pi }{4}\right)\right] \\ {z}_{1}{z}_{2}=8\left[-\frac{\sqrt{2}}{2}+i\left(-\frac{\sqrt{2}}{2}\right)\right] \\ &{z}_{1}{z}_{2}=-4\sqrt{2}-4i\sqrt{2} \end{align}[/latex]. Express [latex]z=3i[/latex] as [latex]r\text{cis}\theta [/latex] in polar form. If [latex]\tan \theta =\frac{5}{12}[/latex], and [latex]\tan \theta =\frac{y}{x}[/latex], we first determine [latex]r=\sqrt{{x}^{2}+{y}^{2}}=\sqrt{{12}^{2}+{5}^{2}}=13\text{. [latex]{z}_{0}=2\left(\cos \left(30^\circ \right)+i\sin \left(30^\circ \right)\right)[/latex], [latex]{z}_{1}=2\left(\cos \left(120^\circ \right)+i\sin \left(120^\circ \right)\right)[/latex], [latex]{z}_{2}=2\left(\cos \left(210^\circ \right)+i\sin \left(210^\circ \right)\right)[/latex], [latex]{z}_{3}=2\left(\cos \left(300^\circ \right)+i\sin \left(300^\circ \right)\right)[/latex], [latex]\begin{gathered}x=r\cos \theta \\ y=r\sin \theta \\ r=\sqrt{{x}^{2}+{y}^{2}} \end{gathered}[/latex], [latex]\begin{align}&z=x+yi \\ &z=r\cos \theta +\left(r\sin \theta \right)i \\ &z=r\left(\cos \theta +i\sin \theta \right) \end{align}[/latex], CC licensed content, Specific attribution, http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. Finding powers of complex numbers is greatly simplified using De Moivre’s Theorem. \\ &{z}^{\frac{1}{3}}=2\left(\cos \left(\frac{14\pi }{9}\right)+i\sin \left(\frac{14\pi }{9}\right)\right)\end{align}[/latex], Remember to find the common denominator to simplify fractions in situations like this one. Writing a complex number in polar form involves the following conversion formulas: [latex]\begin{gathered} x=r\cos \theta \\ y=r\sin \theta \\ r=\sqrt{{x}^{2}+{y}^{2}} \end{gathered}[/latex], [latex]\begin{align}&z=x+yi \\ &z=\left(r\cos \theta \right)+i\left(r\sin \theta \right) \\ &z=r\left(\cos \theta +i\sin \theta \right) \end{align}[/latex]. By … Find more Mathematics widgets in Wolfram|Alpha. The horizontal axis is the real axis and the vertical axis is the imaginary axis. [latex]\begin{align}&r=\sqrt{{x}^{2}+{y}^{2}} \\ &r=\sqrt{{\left(-4\right)}^{2}+\left({4}^{2}\right)} \\ &r=\sqrt{32} \\ &r=4\sqrt{2} \end{align}[/latex]. And then the imaginary parts-- we have a 2i. The modulus of a complex number is also called absolute value. The rectangular form of a complex number is denoted by: In the case of a complex number, r signifies the absolute value or modulus and the angle θ is known as the argument of the complex number. where [latex]n[/latex] is a positive integer. Writing it in polar form, we have to calculate [latex]r[/latex] first. Complex numbers in the form [latex]a+bi[/latex] are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. The polar form of a complex number expresses a number in terms of an angle θ\displaystyle \theta θ and its distance from the origin r\displaystyle rr. In other words, given [latex]z=r\left(\cos \theta +i\sin \theta \right)[/latex], first evaluate the trigonometric functions [latex]\cos \theta [/latex] and [latex]\sin \theta [/latex]. Therefore, if we add the two given complex numbers, we get; Again, to convert the resulting complex number in polar form, we need to find the modulus and argument of the number. [latex]\begin{align}&r=\sqrt{{x}^{2}+{y}^{2}} \\ &r=\sqrt{{\left(1\right)}^{2}+{\left(1\right)}^{2}} \\ &r=\sqrt{2} \end{align}[/latex], Then we find [latex]\theta [/latex]. The exponential number raised to a Complex number is more easily handled when we convert the Complex number to Polar form where is the Real part and is the radius or modulus and is the Imaginary part with as the argument. Explanation: The figure below shows a complex number plotted on the complex plane. Find the angle [latex]\theta [/latex] using the formula: [latex]\begin{align}&\cos \theta =\frac{x}{r} \\ &\cos \theta =\frac{-4}{4\sqrt{2}} \\ &\cos \theta =-\frac{1}{\sqrt{2}} \\ &\theta ={\cos }^{-1}\left(-\frac{1}{\sqrt{2}}\right)=\frac{3\pi }{4} \end{align}[/latex]. So we can write the polar form of a complex number as: x + y j = r ( cos θ + j sin θ) \displaystyle {x}+ {y} {j}= {r} {\left ( \cos {\theta}+ {j}\ \sin {\theta}\right)} x+yj = r(cosθ+ j sinθ) r is the absolute value (or modulus) of the complex number. The polar form of a complex number expresses a number in terms of an angle [latex]\theta [/latex] and its distance from the origin [latex]r[/latex]. From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction. Hence. To find the power of a complex number [latex]{z}^{n}[/latex], raise [latex]r[/latex] to the power [latex]n[/latex], and multiply [latex]\theta [/latex] by [latex]n[/latex]. It measures the distance from the origin to a point in the plane. Then, [latex]z=r\left(\cos \theta +i\sin \theta \right)[/latex]. We use the term modulus to represent the absolute value of a complex number, or the distance from the origin to the point [latex]\left(x,y\right)[/latex]. Let r and θ be polar coordinates of the point P(x, y) that corresponds to a non-zero complex number z = x + iy . We call this the polar form of a complex number.. where [latex]k=0,1,2,3,…,n - 1[/latex]. We're asked to add the complex number 5 plus 2i to the other complex number 3 minus 7i. The argument, in turn, is affected so that it adds himself the same number of times as the potency we are raising. [latex]\begin{align}&{z}^{\frac{1}{3}}={8}^{\frac{1}{3}}\left[\cos \left(\frac{\frac{2\pi }{3}}{3}+\frac{2k\pi }{3}\right)+i\sin \left(\frac{\frac{2\pi }{3}}{3}+\frac{2k\pi }{3}\right)\right] \\ &{z}^{\frac{1}{3}}=2\left[\cos \left(\frac{2\pi }{9}+\frac{2k\pi }{3}\right)+i\sin \left(\frac{2\pi }{9}+\frac{2k\pi }{3}\right)\right] \end{align}[/latex], There will be three roots: [latex]k=0,1,2[/latex]. We use [latex]\theta [/latex] to indicate the angle of direction (just as with polar coordinates). 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To find the potency of a complex number in polar form one simply has to do potency asked by the module. Calculate the new trigonometric expressions and multiply through by [latex]r[/latex]. It is the distance from the origin to the point [latex]\left(x,y\right)[/latex]. It is the standard method used in modern mathematics. We often use the abbreviation [latex]r\text{cis}\theta [/latex] to represent [latex]r\left(\cos \theta +i\sin \theta \right)[/latex]. and the angle θ is given by . Let us find [latex]r[/latex]. Now that we can convert complex numbers to polar form we will learn how to perform operations on complex numbers in polar form. Below is a summary of how we convert a complex number from algebraic to polar form. Example: Find the polar form of complex number 7-5i. Write the complex number in polar form. Polar form is where a complex number is denoted by the length (otherwise known as the magnitude, absolute value, or modulus) and the angle of its vector (usually denoted by … In polar coordinates, the complex number [latex]z=0+4i[/latex] can be written as [latex]z=4\left(\cos \left(\frac{\pi }{2}\right)+i\sin \left(\frac{\pi }{2}\right)\right)[/latex] or [latex]4\text{cis}\left(\frac{\pi }{2}\right)[/latex]. The form z=a+bi is the rectangular form of a complex number. where [latex]r[/latex] is the modulus and [latex]\theta [/latex] is the argument. On the complex plane, the number [latex]z=4i[/latex] is the same as [latex]z=0+4i[/latex]. There are several ways to represent a formula for finding \(n^{th}\) roots of complex numbers in polar form. Then a new complex number is obtained. Therefore, the required complex number is 12.79∠54.1°. Example 1. Entering complex numbers in polar form: But in polar form, the complex numbers are represented as the combination of modulus and argument. Notice that the product calls for multiplying the moduli and adding the angles. For example, the graph of [latex]z=2+4i[/latex], in Figure 2, shows [latex]|z|[/latex]. [latex]\begin{align}&{\left(a+bi\right)}^{n}={r}^{n}\left[\cos \left(n\theta \right)+i\sin \left(n\theta \right)\right]\\ &{\left(1+i\right)}^{5}={\left(\sqrt{2}\right)}^{5}\left[\cos \left(5\cdot \frac{\pi }{4}\right)+i\sin \left(5\cdot \frac{\pi }{4}\right)\right] \\ &{\left(1+i\right)}^{5}=4\sqrt{2}\left[\cos \left(\frac{5\pi }{4}\right)+i\sin \left(\frac{5\pi }{4}\right)\right] \\ &{\left(1+i\right)}^{5}=4\sqrt{2}\left[-\frac{\sqrt{2}}{2}+i\left(-\frac{\sqrt{2}}{2}\right)\right] \\ &{\left(1+i\right)}^{5}=-4 - 4i \end{align}[/latex]. The n th Root Theorem Find the absolute value of [latex]z=\sqrt{5}-i[/latex]. Divide [latex]\frac{{r}_{1}}{{r}_{2}}[/latex]. The quotient of two complex numbers in polar form is the quotient of the two moduli and the difference of the two arguments. So let's add the real parts. Convert the complex number to rectangular form: [latex]z=4\left(\cos \frac{11\pi }{6}+i\sin \frac{11\pi }{6}\right)[/latex]. Finding Roots of Complex Numbers in Polar Form To find the nth root of a complex number in polar form, we use the Root Theorem or De Moivre’s Theorem and raise the complex number to a power with a rational exponent. [latex]{z}_{1}{z}_{2}=-4\sqrt{3};\frac{{z}_{1}}{{z}_{2}}=-\frac{\sqrt{3}}{2}+\frac{3}{2}i[/latex]. Replace [latex]r[/latex] with [latex]\frac{{r}_{1}}{{r}_{2}}[/latex], and replace [latex]\theta [/latex] with [latex]{\theta }_{1}-{\theta }_{2}[/latex]. [latex]\begin{align}&|z|=\sqrt{{x}^{2}+{y}^{2}}\\ &|z|=\sqrt{{\left(\sqrt{5}\right)}^{2}+{\left(-1\right)}^{2}} \\ &|z|=\sqrt{5+1} \\ &|z|=\sqrt{6} \end{align}[/latex]. There are several ways to represent a formula for finding [latex]n\text{th}[/latex] roots of complex numbers in polar form. The polar form of a complex number is a different way to represent a complex number apart from rectangular form. 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