Question 6, Review Exercise

Solutions of Question 6 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Evaluate $\left[\dfrac{1}{i^{10}}+(2-i)^{2}+\sqrt{-25}\right]^{3}$

Solution.

$$\begin{align*} &\left[\dfrac{1}{i^{10}} + (2 - i)^2 + \sqrt{-25}\right]^3\\ =&\left[\dfrac{1}{(i^2)^5} + ( 4 - 4i + i^2) + 5i \right]^3\\ =&\left[\dfrac{1}{(-1)^5} + ( 4 - 4i -1) + 5i \right]^3\\ =&\left[-1 + 3 - 4i + 5i \right]^3\\ =&\left[ 2 + i \right]^3\\ =& 2^3+3(2^2)i+3(2)i^2+i^3\\ =& 8+12i-6-i\\ =& 2+11i\end{align*}$$

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