Algebra 2 Common Core

Published by Prentice Hall
ISBN 10: 0133186024
ISBN 13: 978-0-13318-602-4

Chapter 4 - Quadratic Functions and Equations - 4-8 Complex Numbers - Practice and Problem-Solving Exercises - Page 254: 55



Work Step by Step

Using $(a+b)(c+d)=ac+ad+bc+bd$ or the FOIL Method, the given expression, $ (3+\sqrt{-4})+(4+\sqrt{-1}) ,$ is equivalent to \begin{align*} & 3(4)+3(\sqrt{-1})+\sqrt{-4}(4)+\sqrt{-4}(\sqrt{-1}) \\&= 12+3(\sqrt{-1})+4\sqrt{-4}+\sqrt{-4}(\sqrt{-1}) .\end{align*} Using $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$, the expression above is equivalent to \begin{align*} & 12+3(\sqrt{-1})+4\sqrt{-1}\cdot\sqrt{4}+\sqrt{-1}\cdot\sqrt{4}(\sqrt{-1}) \\&= 12+3(\sqrt{-1})+4\sqrt{-1}\cdot2+\sqrt{-1}\cdot2(\sqrt{-1}) \\&= 12+3(\sqrt{-1})+8\sqrt{-1}+2\sqrt{-1}(\sqrt{-1}) .\end{align*} Since $i=\sqrt{-1},$ the expression above is equivalent to \begin{align*} & 12+3i+8i+2i(i) \\&= 12+3i+8i+2i^2 \\&= 12+3i+8i+2(-1) &\text{ (use $i^2=-1$)} \\&= 12+3i+8i-2 .\end{align*} Combining the real parts and the imaginary parts, the expression above is equivalent to \begin{align*} & (12-2)+(3i+8i) \\&= 10+11i .\end{align*}
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