Write the given numbers in the form a+ib.

6. Write the given numbers in the form $a+ib$.

• a) $3i^5 - i^4 + 7i^3 - 10i^2 - 9$ Recall the powers of $i$: $i^1=i$, $i^2=-1$, $i^3=-i$, $i^4=1$. For $i^5$, we have $i^5 = i^4 \cdot i = 1 \cdot i = i$. Substitute these values into the expression:

  $$3(i) - (1) + 7(-i) - 10(-1) - 9$$ $$= 3i - 1 - 7i + 10 - 9$$

  Group the real and imaginary terms:

  $$= (-1 + 10 - 9) + (3i - 7i)$$ $$= (0) + (-4i)$$ $$= -4i$$

  The expression in the form $a+ib$ is $\boxed{0 - 4i}$.

• b) $2i^6 + \left(\frac{2}{-i}\right)^3 + 5i^5 - 12i$ First, evaluate the powers of $i$: $i^6 = i^4 \cdot i^2 = 1 \cdot (-1) = -1$ $i^5 = i^4 \cdot i = 1 \cdot i = i$

  Next, simplify the term $\left(\frac{2}{-i}\right)^3$:

  $$\frac{2}{-i} = \frac{2}{-i} \cdot \frac{i}{i} = \frac{2i}{-i^2} = \frac{2i}{-(-1)} = \frac{2i}{1} = 2i$$

  Now, cube this result:

  $$(2i)^3 = 2^3 \cdot i^3 = 8 \cdot (-i) = -8i$$

  Substitute all simplified terms back into the original expression:

  $$2(-1) + (-8i) + 5(i) - 12i$$ $$= -2 - 8i + 5i - 12i$$

  Group the real and imaginary terms:

  $$= (-2) + (-8i + 5i - 12i)$$ $$= -2 + (-15i)$$ $$= -2 - 15i$$

  The expression in the form $a+ib$ is $\boxed{-2 - 15i}$.

7. Find the polar coordinate form of the complex numbers.
The polar form of a complex number $z = a+bi$ is $z = r(\cos\theta + i\sin\theta)$, where $r = |z| = \sqrt{a^2+b^2}$ and $\theta = \arg(z)$.

• a) $z = i$ This can be written as $z = 0 + 1i$. Calculate the modulus $r$:

  $$r = \sqrt{0^2 + 1^2} = \sqrt{1} = 1$$

  Calculate the argument $\theta$: The complex number $i$ lies on the positive imaginary axis.

  $$\theta = \frac{\pi}{2}$$

  So, the polar coordinate form is $\boxed{1\left(\cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right)\right)}$.

• b) $z = 3 - 4i$ This is in the form $a+bi$ with $a=3$ and $b=-4$. Calculate the modulus $r$:

  $$r = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

  Calculate the argument $\theta$: The complex number $3-4i$ is in the fourth quadrant.

  $$\theta = \arctan\left(\frac{-4}{3}\right)$$

  Using a calculator, $\theta \approx -0.927$ radians (or approximately $-53.13^\circ$). So, the polar coordinate form is $\boxed{5(\cos(-0.927) + i\sin(-0.927))}$.

8. Find the values of $x$ and $y$ that satisfy the equation $(x+y) + i(x-y) = 14.8 + 6.2i$.

For two complex numbers to be equal, their real parts must be equal and their imaginary parts must be equal.

• Step 1: Equate the real parts. $$x+y = 14.8 \quad (1)$$
• Step 2: Equate the imaginary parts. $$x-y = 6.2 \quad (2)$$
• Step 3: Solve the system of linear equations. Add equation (1) and equation (2): $$(x+y) + (x-y) = 14.8 + 6.2$$ $$2x = 21$$ $$x = \frac{21}{2}$$ $$x = 10.5$$ Substitute the value of $x$ into equation (1): $$10.5 + y = 14.8$$ $$y = 14.8 - 10.5$$ $$y = 4.3$$ The values are $\boxed{x=10.5, y=4.3}$.

9. Express $(-1+i)$ in the form $re^{i\theta}$, where $r$ is positive and $-\pi < \theta < \pi$.

Let $z = -1+i$.

• Step 1: Find the modulus $r$. $$r = |-1+i| = \sqrt{(-1)^2 + (1)^2} = \sqrt{1+1} = \sqrt{2}$$
• Step 2: Find the argument $\theta$. The complex number $-1+i$ corresponds to the point $(-1,1)$ in the complex plane, which is in the second quadrant. The reference angle $\alpha$ is given by $\tan\alpha = \left|\frac{1}{-1}\right| = 1$. So, $\alpha = \frac{\pi}{4}$. Since the point is in the second quadrant, the argument $\theta$ is: $$\theta = \pi - \alpha = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$$ This value of $\theta = \frac{3\pi}{4}$ satisfies the condition $-\pi < \theta < \pi$.
• Step 3: Write in the form $re^{i\theta}$. $$z = \sqrt{2}e^{i\frac{3\pi}{4}}$$ The expression is $\boxed{\sqrt{2}e^{i\frac{3\pi}{4}}}$.