Maharashtra Board Solutions for Chapter: Complex Numbers, Exercise 5: MISCELLANEOUS EXERCISE

Simplify the following and express in the form $a+ib$.

$\frac{3i^5+2i^7+i^9}{i^6+2i^8+3i^{18}}$

Simplify the following and express in the form $a+ib$.

$\frac{5+7i}{4+3i}+\frac{5+7i}{4-3i}$

Evaluate ${\left(1-i+i^2\right)}^{-15}$.

Evaluate $\left(i^{131}+i^{49}\right)$.

Simplify $\frac{i^{29}+i^{39}+i^{49}}{i^{30}+i^{40}+i^{50}}$

Simplify: $\left(i^{65}+\frac{1}{i^{145}}\right)$.

Simplify: $\frac{i^{238}+i^{236}+i^{234}+i^{232}+i^{230}}{i^{228}+i^{226}+i^{224}+i^{222}+i^{220}}$.

If $\omega$ is the cube root of unity, then find the value of ${\left(\frac{-1+i\sqrt{3}}{2}\right)}^{18}+{\left(\frac{-1-i\sqrt{3}}{2}\right)}^{18}$.