Fill in the blanks. When evaluating limits at infinity for complicated rational functions, you can divide the numerator and denominator by the ____ te

Find the distance between the complex numbers in the complex plane. ${\displaystyle 1+4i,-1+7i}$

Find roots of a complex variable such that ${\displaystyle 1+{\omega }^m+{\omega }^{2m}+\cdots +{\omega }^{(n-1)m}=0}$
I need some help understanding the intuition behind the following question:
Consider the root of ${\displaystyle z^n=1}$ given by ${\displaystyle \omega =\cos \frac{2\pi }{n}+i\sin \frac{2\pi }{n}}$ . For which integers m is ${\displaystyle 1+{\omega }^m+{\omega }^{2m}+\cdots +{\omega }^{(n-1)m}=0}$?

Complex number in rectangular form What is (1+2j) + (1+3j)? Your answer should contain three significant figures.

Find a subgroup of order ${\displaystyle 6\in U\left(700\right)}$.

If m be the least value of ${\displaystyle {|z-3+4i|}^2+{|z-5-2i|}^2,z}$ belongs to C, attained at ${\displaystyle z=z_0}$ then what does the ordered pair ${\displaystyle (\left|z_0\right|,m)}$ is equal to?

What is 2+i plus 2+4i?

Multiply: $$(4+5i)\times (6+2i)$$.