- Under a power, treat the modulus and argument separately. - De Moivre’s theorem changes the argument by multiplying it by the exponent.

1. Square the modulus: \(2^2=4\). 2. Multiply the argument by \(2\): \(2\cdot\frac{\pi}{3}=\frac{2\pi}{3}\). 3. Therefore, \(z^2=4e^{i2\pi/3}\).

\(z^2=4e^{i2\pi/3}\)

- Look for all numbers whose square is the given positive number. - Think of the directions to a positive and a negative real number on the real axis.

1. The square roots of \(9\) are \(3\) and \(-3\). 2. These points are opposite each other on the real axis, so their rays differ by \(180^\circ\).

Two square roots; angle between them \(180^\circ\).

- Think about which numbers can have a positive integer power equal to zero. - Can the origin determine a direction in the complex plane?

1. If a complex number has fifth power zero, its modulus raised to the fifth power is zero, so its modulus is zero. 2. Therefore, \(w=0\) is the only solution. 3. The zero complex number has no direction from the origin, so its argument is undefined.

\(w=0\) only; one distinct solution; its argument is undefined.

- Find the distance from the origin and choose the angle in the stated interval. - Powers multiply the argument by the exponent. - A complex number is real when it lies on the horizontal axis. - Use the value of \(z^4\) to find \(z^8\).

1. \(|z|=\sqrt{2}\) and the argument in \([0,2\pi)\) is \(\frac{\pi}{4}\), so \(z=\sqrt{2}e^{i\pi/4}\). 2. \(z^n=(\sqrt{2})^ne^{in\pi/4}\) is real when \(\frac{n\pi}{4}\) is an integer multiple of \(\pi\). The smallest positive value is \(n=4\), and \(z^4=4e^{i\pi}=-4\). 3. \(z^8=(z^4)^2=16\), which is a positive integer.

1. \(z=\sqrt{2}e^{i\pi/4}\) 2. \(n=4\) and \(z^4=-4\) 3. \(z^8=16\)

- The fourth roots all have the same modulus. - Their arguments are equally spaced around a full circle. - Start from an argument of the original negative real number, then divide the complete family of arguments by \(4\).

1. Write \(-16=16e^{i\pi}\). The fourth roots have modulus \(2\) and arguments \(\frac{\pi+2\pi k}{4}\) for \(k=0,1,2,3\). 2. The four arguments are \(\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}\). 3. Converting to rectangular form gives \(\sqrt{2}+i\sqrt{2}\), \(-\sqrt{2}+i\sqrt{2}\), \(-\sqrt{2}-i\sqrt{2}\), and \(\sqrt{2}-i\sqrt{2}\).

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

- The root index determines how many distinct roots a nonzero complex number has. - Treat the modulus separately from the arguments. - Equal spacing divides one full revolution into equal parts. - Use the axis labels in the diagram to identify the root requested in part d.

1. A nonzero complex number has \(5\) distinct fifth roots. 2. If \(|z|^5=|32i|=32\), then \(|z|=\sqrt[5]{32}=2\). 3. The roots are equally spaced around a full circle, so the angular spacing is \(\frac{2\pi}{5}\). 4. The root on the positive imaginary axis is plotted at \((0, 2)\), and the diagram labels it \(B\).

a) \(5\) b) \(2\) c) \(\frac{2\pi}{5}\) d) Point \(B\)

- Begin with the length and direction of the point \(1\). - Compare the arrows from the origin before and after the first multiplication. What stays unchanged?

1. After one multiplication, the point is \(i\), with argument \(90^\circ\). 2. After two multiplications, the point is \(-1\), with argument \(180^\circ\). 3. After three multiplications, the point is \(-i\), with argument \(270^\circ\). 4. Each step is a \(90^\circ\) counterclockwise turn with no change in length. Thus \(i^3=-i\).

After step 1: \(i\), \(90^\circ\). After step 2: \(-1\), \(180^\circ\). After step 3: \(-i\), \(270^\circ\). Each step turns \(90^\circ\) counterclockwise; \(i^3=-i\).

- What happens to the argument when a number is cubed? - Which different root angles have triples that point in the same direction as the original number? - Match the expected directions with the labeled points.

1. A cube root has modulus \(\sqrt[3]{8}=2\). 2. Its argument must satisfy \(3\theta=360^\circ k\), giving distinct angles \(0^\circ,120^\circ,240^\circ\). 3. The corresponding points are \((2,0)\), \((-1,\sqrt{3})\), and \((-1,-\sqrt{3})\), labeled \(A\), \(B\), and \(D\). Point \(C\) is not a cube root.

Points \(A\), \(B\), and \(D\); arguments \(0^\circ,120^\circ,240^\circ\).

- Work backward from what squaring does to a number's distance from the origin. - Could two different directions lead to the same direction after doubling their angles? - Use the given argument interval when you write both roots.

1. Each square root has modulus \(\sqrt{9}=3\). 2. To square to an argument of \(\frac{4\pi}{5}\), a root may have argument \(\frac{2\pi}{5}\). Adding \(\pi\) gives the other argument \(\frac{7\pi}{5}\). 3. The roots are \(3e^{i2\pi/5}\) and \(3e^{i7\pi/5}\). Their squared arguments are \(\frac{4\pi}{5}\) and \(\frac{14\pi}{5}=\frac{4\pi}{5}+2\pi\), which represent the same direction.

\(3e^{i2\pi/5}\) and \(3e^{i7\pi/5}\). Both square to \(9e^{i4\pi/5}\) because their squared arguments differ by \(2\pi\).

- First express the number's modulus and direction in exponential form. - How are the arguments of a number and its square related? - How can two different arguments lead to the same angle after squaring?

1. Write \(4i=4e^{i\pi/2}\). Each square root has modulus \(\sqrt{4}=2\). 2. The arguments are \(\frac{\pi}{4}\) and \(\frac{5\pi}{4}\), a half-turn apart. The roots in exponential form are \(2e^{i\pi/4}\) and \(2e^{i5\pi/4}\). 3. Converting gives \(2(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4})=\sqrt{2}+\sqrt{2}i\) and its opposite \(-\sqrt{2}-\sqrt{2}i\).

Exponential polar: \(2e^{i\pi/4}\) and \(2e^{i5\pi/4}\). Rectangular: \(\sqrt{2}+\sqrt{2}i\) and \(-\sqrt{2}-\sqrt{2}i\).

- Look for a pattern the roots of one fixed cube equation must share. - Imagine cubing the points in each diagram. Would their resulting lengths and directions agree?

1. All cube roots of the same nonzero number must have the same modulus and successive arguments \(\frac{2\pi}{3}=120^\circ\) apart. 2. In a), the three points have radius \(2\) and arguments \(90^\circ\), \(210^\circ\), and \(330^\circ\), so this is possible. 3. In b), the three points have radius \(2\), but their arguments are \(0^\circ\), \(90^\circ\), and \(180^\circ\). The gaps are not equal, so it is not a set of three cube roots.

Diagram a) only: all three points have equal radius and angular gaps of \(120^\circ\). Diagram b) has unequal angular gaps.

- Check the proposed root rather than assuming it is wrong. - Could another number have the same square as the one the student found? - How could you show that your list is complete, rather than only that its entries work?

1. Squaring gives \((1+i)^2=1+2i+i^2=2i\), so the student's proposed root is correct. 2. The opposite number \(-1-i\) has the same square, so it is the missing root. 3. A nonzero complex number has exactly two distinct square roots, so the list is complete.

The claim is incomplete. Both \(1+i\) and \(-1-i\) square to \(2i\); these are all two square roots.

- Under an integer power, what happens separately to a complex number's length and direction? - After computing the angle, remove any whole revolutions. - Convert the final angle to real and imaginary components.

1. Cubing multiplies the angle by \(3\) and raises the modulus to the third power, giving \(z^3=8e^{i5\pi/2}\). 2. Since \(\frac{5\pi}{2}=\frac{\pi}{2}+2\pi\), the required argument is \(\frac{\pi}{2}\). 3. Thus \(z^3=8e^{i\pi/2}=8i\).

Exponential polar: \(8e^{i\pi/2}\); rectangular: \(8i\)

- Find the modulus as the distance from the origin. - Use the quadrant to determine the argument. - Apply De Moivre’s theorem to powers. - A complex number is real when its sine component is zero.

1. The modulus is \(2\). Since the point lies in Quadrant II with reference angle \(\frac{\pi}{6}\), its argument is \(\frac{5\pi}{6}\). Thus, \(z=2e^{i5\pi/6}\). 2. \(z^6=2^6e^{i5\pi}=64(\cos5\pi+i\sin5\pi)=-64\). 3. \(z^n=2^ne^{i5n\pi/6}\) is real when \(\sin\left(\frac{5n\pi}{6}\right)=0\). Thus, \(\frac{5n}{6}\) must be an integer. Since \(5\) and \(6\) are relatively prime, \(n\) must be a positive multiple of \(6\).

1. \(z=2e^{i5\pi/6}\) 2. \(z^6=-64\) 3. \(n=6k\), where \(k\) is a positive integer

- An equation \(z^n=a\) has \(n\) complex roots. - The roots are equally spaced around a circle. - Find the roots in polar form, then convert to rectangular form. - Use conjugate factors when multiplying the nonreal roots.

1. Write \(8=8e^{i0}\). The cube roots are \(z_k=2e^{i2k\pi/3}\) for \(k=0, 1, 2\). 2. Therefore, \(z_0=2\), \(z_1=-1+i\sqrt{3}\), and \(z_2=-1-i\sqrt{3}\). 3. Their sum is \(2+(-1+i\sqrt{3})+(-1-i\sqrt{3})=0\). 4. Their product is \(2(-1+i\sqrt{3})(-1-i\sqrt{3})=2\cdot(1+3)=8\).

1. \(z_0=2\), \(z_1=-1+i\sqrt{3}\), \(z_2=-1-i\sqrt{3}\) 2. \(z_0+z_1+z_2=0\) 3. \(z_0z_1z_2=8\)

- Isolate \(z^3\) before converting the right side to polar form. - Use all coterminal arguments of the original complex number before dividing by the root index. - The three cube-root arguments should be equally spaced around the circle.

1. Rewrite the equation as \(z^3=-8i\). 2. In polar form with a radian argument in \([0,2\pi)\), \(-8i=8e^{i3\pi/2}\). 3. The cube roots have modulus \(2\) and arguments \(\frac{3\pi/2+2\pi k}{3}\) for \(k=0,1,2\). These are \(\frac{\pi}{2}\), \(\frac{7\pi}{6}\), and \(\frac{11\pi}{6}\). 4. Converting to rectangular form gives \(2i\), \(-\sqrt{3}-i\), and \(\sqrt{3}-i\).

\(z\in\{2i, -\sqrt{3}-i, \sqrt{3}-i\}\)

- What are the length and direction of \(3+4i\) in the complex plane? - How does squaring a complex number change its modulus and argument? - How can the modulus and argument of a root give its real and imaginary parts?

1. The modulus of \(3+4i\) is \(5\). Its argument is \(\theta=\arctan\left(\frac{4}{3}\right)\) in Quadrant I, so \(3+4i=5e^{i\theta}\). 2. A square root has modulus \(\sqrt{5}\). The two root arguments in \([0,2\pi)\) are \(\frac{\theta}{2}\) and \(\frac{\theta}{2}+\pi\). 3. Since \(\cos\theta=\frac{3}{5}\) and \(\frac{\theta}{2}\) is in Quadrant I, \(\cos\frac{\theta}{2}=\sqrt{\frac{1+3/5}{2}}=\frac{2}{\sqrt{5}}\) and \(\sin\frac{\theta}{2}=\sqrt{\frac{1-3/5}{2}}=\frac{1}{\sqrt{5}}\). 4. The roots are \(\sqrt{5}\left(\cos\frac{\theta}{2}+i\sin\frac{\theta}{2}\right)=2+i\) and its negative, \(-2-i\). 5. Therefore, \((x, y)=(2, 1)\) or \((x, y)=(-2, -1)\). Both roots square to \(3+4i\).

Root modulus: \(\sqrt{5}\). Root arguments: \(\frac{\theta}{2}\) and \(\frac{\theta}{2}+\pi\), where \(\theta=\arctan\left(\frac{4}{3}\right)\). Pairs: \((2, 1)\) and \((-2, -1)\).

- Check whether the three reported angles actually determine three different directions. - Distinct roots come from different coterminal arguments of the original number before dividing the angle by the root index. - The arguments of three cube roots should be evenly spaced around the circle.

1. The three reported angles differ by whole multiples of \(2\pi\), so all three expressions represent the same complex number. Mateo added full turns after taking the cube root instead of using different arguments of \(8i\) before dividing by \(3\). 2. Write the arguments of \(8i\) as \(\frac{\pi}{2}+2\pi k\). Dividing by \(3\) gives root arguments \(\frac{\pi}{6}+\frac{2\pi k}{3}\). 3. For \(k=0,1,2\), the distinct arguments are \(\frac{\pi}{6}\), \(\frac{5\pi}{6}\), and \(\frac{3\pi}{2}\).

The reported roots are not distinct because their arguments differ by multiples of \(2\pi\). The three roots are \(2e^{i\pi/6}\), \(2e^{i5\pi/6}\), and \(2e^{i3\pi/2}\).

- Take the root of the modulus separately from the arguments. - Include the full family of coterminal arguments of the original complex number before dividing by the root index. - The three final root arguments should be equally spaced.

1. Each cube root has modulus \(\sqrt[3]{8}=2\). 2. The arguments satisfy \(3\theta=\frac{\pi}{4}+2\pi k\), so \(\theta=\frac{\pi}{12}+\frac{2\pi k}{3}\). 3. For \(k=0,1,2\), the arguments are \(\frac{\pi}{12}\), \(\frac{3\pi}{4}\), and \(\frac{17\pi}{12}\). 4. These three arguments lie in \([0,2\pi)\) and differ by \(2\pi/3\), so they give all three distinct roots.

\(z=2e^{i\pi/12},\ 2e^{i3\pi/4},\ 2e^{i17\pi/12}\)

- Work forward from the known root to recover the original number. - Polar form makes the modulus and angle changes under cubing easy to track. - What pattern should all cube roots of one nonzero number make in the complex plane?

1. Since \(u^3=z\), compute \((1+i)^2=2i\) and then \((1+i)^3=2i(1+i)=-2+2i\). 2. The modulus of \(u\) is \(\sqrt{2}\) and its argument is \(\frac{\pi}{4}\). Cubing gives \(|z|=(\sqrt{2})^3=2\sqrt{2}\) and \(\arg(z)=\frac{3\pi}{4}\). 3. The three cube-root arguments are equally spaced by \(\frac{2\pi}{3}\). Starting from \(\frac{\pi}{4}\), the other two are \(\frac{11\pi}{12}\) and \(\frac{19\pi}{12}\).

a) \(z=-2+2i=2\sqrt{2}e^{i3\pi/4}\) b) \(\frac{11\pi}{12}\) and \(\frac{19\pi}{12}\)

- Translate “negative real” into a condition on the argument of the fourth power. - Work backward from the possible fourth-power arguments before using the quadrant condition. - The modulus of a fourth power is the fourth power of the original modulus.

1. Let the argument of \(z\) be \(\theta\). For \(z^4\) to be negative real, \(4\theta\equiv\pi\pmod{2\pi}\). 2. Thus \(\theta=\frac{\pi}{4}+\frac{k\pi}{2}\). The only such angle in Quadrant II is \(\theta=\frac{3\pi}{4}\). 3. Therefore, \(z=2\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right)=-\sqrt{2}+i\sqrt{2}\). 4. Its fourth power has modulus \(2^4=16\) and argument \(4\cdot\frac{3\pi}{4}=3\pi\), so \(z^4=-16\).

\(z=-\sqrt{2}+i\sqrt{2}\) and \(z^4=-16\)