Question 1

What is the value of $i^{35}$?

Accepted Answer

To find the value of $i^{35}$, we divide the exponent 35 by 4. $35 = 4 	imes 8 + 3$. Therefore, $i^{35} = i^{4 	imes 8 + 3} = (i^4)^8 \cdot i^3$. Since $i^4 = 1$ and $i^3 = -i$, we get $i^{35} = (1)^8 \cdot (-i) = -i$.

Question 2

What is the multiplicative inverse of $2 - 3i$?

Accepted Answer

The multiplicative inverse of a complex number $z = a + ib$ is given by $\bar{z} / |z|^2$. For $z = 2 - 3i$, the conjugate is $\bar{z} = 2 + 3i$ and $|z|^2 = 2^2 + (-3)^2 = 4 + 9 = 13$. Therefore, the multiplicative inverse is $(2 + 3i) / 13$.

Question 3

What is the modulus of the complex number $-1 + i\sqrt{3}$?

Accepted Answer

The modulus of a complex number $z = x + iy$ is given by $|z| = \sqrt{x^2 + y^2}$. Here, $x = -1$ and $y = \sqrt{3}$. Thus, $|z| = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$.

Question 4

What is the conjugate of the complex number $3 - 4i$?

Accepted Answer

The conjugate of a complex number $z = a + ib$ is $\bar{z} = a - ib$. For $z = 3 - 4i$, changing the sign of the imaginary part gives $3 + 4i$.

Question 5

What is the polar form of the complex number $1 + i$?

Accepted Answer

For $z = 1 + i$, $r = |z| = \sqrt{1^2 + 1^2} = \sqrt{2}$. The argument $	heta = 	an^{-1}(1/1) = \pi/4$. Thus, the polar form is $r(\cos	heta + i\sin	heta) = \sqrt{2}(\cos(\pi/4) + i\sin(\pi/4))$.

Complex Numbers and Quadratic Equations