1.2 Complex numbers

Evaluate the following numbers.

(a) [math]3i-2i^3[/math]

(b) [math]3-i+(4+2i)[/math]

(c) [math]\dfrac{2}{i}+\dfrac{i}{2}[/math]

(d) [math](3-i)(4+5i)[/math]

If [math]\sqrt{-2}\cdot(-\sqrt{-12})=a\sqrt{b}[/math] where [math]a[/math] and [math]b[/math] are real numbers, what is the value of [math]a[/math] and [math]b[/math]?

If [math]\dfrac{\sqrt{6}}{\sqrt{-2}-\sqrt{-8}}=\sqrt{a}i[/math] where [math]a[/math]is a real number, what is the value of [math]a[/math]?

If [math](\sqrt{-3}-\sqrt{2})^2=a+b\sqrt{c}i[/math] where [math]a, b[/math] and [math]c[/math] are real numbers, what are the values of [math]a, b[/math] and [math]c[/math]?

Evaluate the following numbers.

(a) [math]i^{3000}+i^{-3000}[/math]

(b) [math]5-6i-7+9i[/math]

(c) [math](i+2)(i^2+4)[/math]

(d) [math](40+40i)(20+20i)[/math]

If [math]\dfrac{\sqrt{3}}{\sqrt{-12}}=ai[/math] where [math]a[/math] is a real number, what is the value of [math]a[/math]?

If [math](\sqrt{-3}-\sqrt{2})^2=-\sqrt{a}+\sqrt{b}i[/math] where [math]a[/math] and [math]b[/math] are real numbers, what are the values of [math]a[/math] and [math]b[/math]?

If [math](\sqrt{-3}-\sqrt{2})^2=a+bi[/math] where [math]a[/math] and [math]b[/math] are real numbers, what is the value of [math]a[/math]?

If [math]A=x+(y+5)i[/math] and [math]B=9-y+xi[/math] where [math]x, y\in\mathbb{R}[/math] and [math]A=B[/math], what is the value of [math]x+y[/math]?

Let [math]C=2+a-2ai+7i[/math] where [math]a[/math] is a constant.

(a) If [math]a\in\mathbb{R}[/math] and [math]C[/math] is a pure imaginary number, what is the value of [math]a[/math]?

(b) If [math]a[/math] is a pure imaginary number and [math]C[/math] is a real number, what is the value of [math]-ai[/math]?

Consider four numbers: $$a=(1-2i)^2, b=-\sqrt{-3}\sqrt{-5}, c=\dfrac{32}{i^7}, d=(4-3i)(4+3i)$$If there are [math]x[/math] real numbers and [math]y[/math] pure imaginary numbers, what are [math]x[/math] and [math]y[/math]?

If [math]A=x-y+5i[/math] and [math]B=4+(2x+y)i[/math] where [math]x, y\in\mathbb{R}[/math] and [math]A=B[/math], what is the value of [math]x+y[/math]?

If [math](1-i)^{2017}=2^a(b+ci)[/math] where [math]a, b[/math], and [math]c[/math] are constants, what is the value of [math]a+b+c[/math]?

Which of the following equations is/are False?

[math]i^{1973}=[/math]

Which of the following is the reciprocal of [math]3i[/math]?